^r 


Mo 
Division 
Range 
Shelf. 
Received  . 


1870 


UNITY  OF  PURPOSE, 


OR 


RATIONAL   ANALYSIS: 


BEI 


A    TREATISE 


DESIGNED 


TO    DISCLOSE   PHYSICAL   TRUTHS,   AND   TO    DETECT 
AND    EXPOSE    POPULAR   ERRORS. 


BY    AUGUSTUS    YOUNG. 


"  These  are  not,  perhaps,  very  attractive  speculations ;  they  disturb  old  and  favorite 
associations ;  they  serve  to  reduce  many  cherished  traditions,  much  painfully  acquired 
knowledge,  to  obsolete  lore  ;  but  these  things  are  so,  and  we  must  accustom  ourselves  to 
regard  them  and  their  consequences  without  shrinking." 


BOSTON: 

PRINTED  BY  S.  N.   DICKINSON  &   CO. 
1846. 


Entered  according  to  Act  of  Congress,  in  the  year  1846,  by 

AUGUSTUS   YOUNG, 
In  the  Clerk's  Office  of  the  District  Court  of  the  District  of  Vermont. 


CONTENTS. 


[Although  in  the  work,  some  of  the  respective  Chapters  are  divided  into,  or  are 
composed  of  several  Sections,  such  division  will  be  disregarded  in  the  Table  of 
Contents, — the  subject  matters  of  the  Chapters  only  being  given.] 

Page. 

Introduction, 5 

CHAPTER    I. 
On  the  Quadrature  of  the  Circle, 27 

CHAPTER    II. 
On  the  Law  of  Gravity,  and  the  Popular  Deduction  and  Promulgation  of  the 

Supposed  Law, 98 

CHAPTER    III. 

Of  the  Proper  Elements  from  which  to  determine  the  Laws  of  Force  and 
Motion,  incident  to  the  Heavenly  Bodies,  in  their  Eternal  Rounds, 198 

CHAPTER    IV. 
On  the  Hypothesis  suggested  by  Kepler  in  respect  to  Elliptical  Orbits, 217 

CHAPTER    V. 
Outline  of  a  Theory  of  the  Solar  System, 231 

CHAPTER   VI. 
On  the  Tides, 246 

CHAPTER    VII. 
An  Investigation  of  the  Theory  of  the  Progressive  Motion  of  Light, 261 


ERRATA. 

Page  37,  line  33,  for  may  read  might. 

Page  48,  line  35,  before  the  word  propose,  read  do  not. 

Page  48,  line  42,  for  divisions,  read  divisors. 

Page  58,  line  43,  before  the  word  will,  read  when  equated. 

Page  60,  line  19,  before  the  word  constantly,  read  when  equated^  will. 

Page  97,  line  11,  for  space,  read  spaces. 

Page  100,  line  37,  erase  the  word  same  commencing  the  line. 

Page  107,  line  14,  for  form,  read  force. 

Page  116,  line  33,  erase  the  words  the  square  root  of. 

Page  117,  line  3,  for  minutes,  read  seconds, 

Page  131,  line  25,  read  the  word  we  after  the  word  it,  at  the  end  of  the  line. 

Page  131,  line  27.  erase  the  word  we  commencing  the  line. 

Page  210,  line  6,  for  or,  read  is. 

Page  216,  line  10,  for  time,  read  line. 

Page  224,  line  30,  for  distance,  read  diameter. 

Page  264,  line  16,  for  sutor,  read  sector. 

Page  286,  line  17,  for  doctrines,  read  dictums. 


INTRODUCTION. 


THE  object  of  this  work  as  declared  in  the  title  page,  is  to 
elicit  inquiry  into,  or  reexamination  of  certain  matters  which  the 
world  at  large  now  receive  implicitly,  at  the  hands  of  others,  as 
scientific  truths ;  but  which,  perhaps,  upon  a  more  rigid  and 
scrupulous  examination,  divested  of  a  too  easy  faith  in  matters 
purely  scientific,  may  be  found  to  be  but  popular  errors,  which 
should  be  eradicated  for  the  benefit  of  physical  science  ;  and 
should  the  fact  be  disclosed  that  many  matters  which  are  now 
almost  universally  esteemed  sublime  scientific  truths,  are  but 
dark  and  occult  errors,  the  inquiry  will  naturally  suggest  itself, 
whether  it  may  not  have  been  found  necessary  to  clothe  them  in 
a  mathematical  dress  so  wholly  incomprehensible  to  the  mass  of 
mankind,  as  to  make  it  a  hopeless  task  from  want  of  leisure  and 
other  facilities,  to  investigate  the  truth  or  falsehood  promulgated 
by  the  learned  through  the  medium  of  what  is  so  triumphantly 
termed  the  higher  branches  of  mathematics  ;  and  hence,  whether 
mankind  in  general,  have  not  been  compelled  to  remain  ignorant 
of  those  supposed  physical  truths,  except  by  a  confidential  faith 
in  those  who  profess  to  teach  those  things.  And  should  my 
labors  in  anywise  serve  to  induce  the  learned  to  stoop  a  little 
more  to  the  necessities  of  the  multitude,  who  lack  leisure  and 
opportunity  to  acquaint  themselves  with  all  the  modern  devices 
of  mathematical  science,  my  object  will  be  fully  accomplished. 

I  am  aware  that  many  will  consider  the  subject  matters  of 
which  I  treat,  (namely,  the  cold,  material  laws  of  nature,)  as 
altogether  grovelling  in  comparison  with  those  flights  of  human 
imagination,  which  flood  the  world  in  such  profusion,  and  which 
so  liberally  dispense  to  mankind  the  benefits  of  modern  literature 
by  means  of  novels  and  tales  of  fiction ;  and  which  are  conceiv- 
ed by  many  to  be  paramount  to  every  thing  intellectual  beside : 
but  to  fulfil  a  task  so  desirable  to  many,  I  confess  I  was  never 


6  INTRODUCTION. 

properly  fitted  and  prepared;  the  preparation  for  it  must,  in 
general,  come  through  the  medium  of  a  collegiate  education  and 
discipline.  Besides,  I  have  ever  looked  upon  the  universe  as  a 
reality,  and  not  as  a  fiction  ;  nor  am  I  without  hope,  that  eternal 
truths  may  yet  become  as  pleasing  to  a  great  portion  of  mankind 
as  is  human  error.  Such  is  certainly  the  case  to  a  very  great 
extent  in  the  moral  world,  and  it  is  certainly  too  manifest  to  be 
overlooked  or  disregarded,  that  if  a  proper  regard  were  had  to 
that  unity  of  purpose  in  which  the  Christian  religion  is  founded, 
the  world  of  mankind  might  be  much  more  readily  or  easily 
squared  and  quadrated  into  what  is  befitting  a  moral  world, 
than  they  could  be  equated  to  the  same  by  means  of  Jewish 
rites  and  ceremonies,  or  by  heathen  philosophy.  It  would 
seem  then  to  be  but  following  out  the  principles  of  Christian 
philsosophy,  to  take  example  from  that  great  dispensation  of 
Divine  truth  in  other  departments  of  science,  or  in  the  natural 
world,  by  endeavoring  to  adopt  that  unity  of  purpose  which 
alone  seems  calculated  to  lead  to  those  truths  which  are  to 
benefit  mankind.  If  such,  then,  were  our  foundation,  —  whatever 
errors  and  mistakes  might  occur  in  our  attempts  to  extend  its 
benefits,  —  we  should  not  be  left  without  a  land-mark,  a  witness, 
contained  in  such  unity  of  purpose,  whereby  to  correct  those 
errors. 

Shall  we  not.  then,  in  our  researches  for  truth,  endeavor  to 
follow  out  the  true  principles  of  Christian  philosophy,  with  a 
view  to  some  unity  of  purpose,  —  as  being  the  most  probable 
condition  upon  which  we  may  hope  for  success? 

In  my  pursuit  after  truth,  I  shall  no  more  be  satisfied  with 
taking  a  middle  flight,  than  Milton  was  in  the  sublimation  of 
fiction  ;  and  as  this  must  often  bring  me  in  contact  with  those 
who  have  attempted  the  most  daring  flights  in  the  scientific 
world,  (especially  in  the  departments  of  mathematics  and  astron- 
omy,) I  shall  speak  of  them  and  their  doings,  with  a  freedom 
and  boldness  which  I  suppose  to  be  justly  due  to  the  subject, 
and  to  mankind  ;  nor  will  I,  in  this  respect,  ask  from  the  world 
other  mercy  than  such  as  I  show  to  others. 

It  may  surprise  the  admiration  which  many  have  been  taught 
to  entertain  of  Sir  Isaac  Newton,  —  who  by  his  flattering  biog- 
raphers, has  been  styled  the  creator  of  natural  philosophy, —~ 
that  I  should  place  Kepler  on  far  higher  ground  as  a  philoso- 
pher, so  far,  at  least,  as  science  has  been  benefited  by  their 
researches  or  hypotheses.  It  may  surprise  many  that  I  should 
esteem  Kepler  as  the  father  of  physical  astronomy ;  and  assert 
that  he  has  done  more  in  laying  the  foundation  of  that  sublime 
science  in  those  eternal  truths  upon  which  its  superstructure 


INTRODUCTION.  7 

should  be  erected,  than  all  others ;  and  that,  had  a  like  sagacity 
succeeded  him  in  those  who  attempted  its  superstructure,  its 
strength  and  beauty  would  have  been  grand  indeed  in  compari- 
son with  its  present  deformity.  And  although  I  shall  treat  of 
Kepler's  elliptical  hypothesis  with  something  more  than  doubts, 
nevertheless,  if  it  were  an  error  in  him,  it  could  not  have  been 
the  cause  of  errors  committed  by  others  in  respect  to  the  laws  of 
force  and  motion,  by  which  the  phenomena  of  the  heavenly 
bodies  can  only  be  explained. 

Sir  Isaac  Newton  lived  in  a  day  of  problems,  —  in  a  time  of 
great  anxiety  in  respect  to  the  science  of  astronomy,  which  had 
recently  been  ushered  forth  to  the  world,  and  somewhat  advanced 
by  the  labors  and  researches  of  Copernicus,  Tycho,  Kepler  and 
Galileo ;  and  hence,  the  world  then  required  some  master  spirit 
to  collect  the  disjointed  parts,  and  to  raise  the  superstructure  to 
their  implicit  admiration  and  content.  Newton  also  lived  in  a 
country  where  science  and  the  arts  were  better  patronized  by 
national  pride  than  in  any  other  country  at  the  time  ;  and  he 
seems  to  have  been  fully  able  to  avail  himself  of  all  advantages 
which  came  in  his  way  ;  for,  if  we  are  to  believe  his  admiring 
biographers,  he  was  often  enabled  to  keep  his  own  discoveries  a 
profound  secret  until  they  were  disclosed  to  the  world  by  others, 
although  he  never  failed  to  assort  his  claim  as  the  discoverer,  and 
to  have  his  claim  allowed  by  his  country ;  as  in  the  case  of  Kep- 
ler, Liebnitz  and  Flamstead ;  although  modern  disclosures  in 
respect  to  the  treatment  of  Flamstead,  have  caused  even  high 
toned  Englishmen  to  declare,  that  the  name  of  Newton  was  no 
longer  to  be  revered.  And  whether  the  doubt  expressed  in  respect 
to  Sir  Richard  Arkwright,  may  not  with  equal  propriety  be  ap- 
plied to  Sir  Isaac  Newton,  is  for  a  candid  world  to  determine, 
namely,  "  some  esteem  this  remarkable  man  as  a  genius  of  the 
first  order,  gifted  with  an  extraordinary  power  of  invention ; 
others,  as  an  artful  contriver,  who  understood  how  to  appropriate 
to  himself  the  discoveries  of  others." 

But  the  biographer  of  Newton  was  certainly  acquainted  with 
what  would  induce  implicit  faith,  and  flatter  a  nation's  vanity, 
when  he  saw  fit  to  insert  the  pitiful  story  about  the  fall  of  the 
apple  from  the  apple-tree,  when  Newton  \vas  sitting  alone  under 
the  tree ;  and  what  Sir  Isaac  Newton  asked  himself  on  that 
occasion ;  and  also  what  he  said  to  his  pet  dog  Diamond,  on  a 
certain  occasion ;  —  how  the  fall  of  the  apple  suggested  to  the 
mind  of  Sir  Isaac,  the  hypothesis  of  the  universal  gravity  of 
inert  matter;  —  how  Sir  Isaac  Newton  devised  a  method  for 
ascertaining  whether  gravity  extended  from  the  earth  to  the  moon, 
and  what  would  be  the  law  of  gravity  in  respect  to  distance  ; — 


8  INTRODUCTION. 

and  how  his  ecstatic  agony,  on  supposing  his  conjecture  about 
to  be  realized  by  proof,  became  so  great  as  to  deprive  him  of  the 
power  to  finish  out  the  problem  and  final  result,  which  was 
thenceforth  to  become  the  great  physical  law  of  the  universe ; 
and  was  therefore  compelled  to  submit  the  finishing  stroke  to 
another  person. 

But  such  stuff  cannot  well  be  accounted  for  upon  any  other 
principle  than  that  of  a  design  to  clothe  Sir  Isaac  Newton  with 
something  superhuman  in  the  eyes  of  the  world,  and  make  him 
an  object  of  adoration  ;  and  thereby  to  induce  a  belief,  that  such 
wonders  could  have  been  performed  by  no  other.  And  such 
efforts  have  had  their  effect ;  Newton  has  been  immortalized, 
and  national  pride  has  been  gratified,  however  true  science  may 
have  been  retarded  in  consequence. 

The  world  has  generally  supposed  that  Newton  first  disclosed 
the  idea  of  attraction  among  the  heavenly  bodies ;  and  that 
such  attraction  was  the  cause  of  the  tides ;  notwithstanding,  Kep- 
ler, some  forty  years  previous  to  the  advent  of  Newton's  apple, 
had  not  only  fully  declared  the  principle  to  the  world,  but  had 
done  it  in  such  a  way  and  manner,  as  not  to  lead  mankind  into 
those  atheistical  and  abhorrent  notions,  which  are  taught  in  the 
Newtonian  philosophy  in  respect  to  the  hypothesis.  Kepler  had 
declared  that  the  tides  were  caused  by  the  attraction  of  the  moon  ; 
that  the  earth  and  moon  mutually  attract  each  other ;  and  that, 
in .  consequence,  they  would  come  together,  and  that  too,  at  a 
point  as  much  nearer  the  earth,  as  the  earth  is  greater  than  the 
moon,  were  they  not  prevented  by  the  motion  of  the  moon  in  its 
orbit.  And  a  better  development  or  declaration  for  the  future 
progress  of  science,  could  not  have  been  made.  It  was  in  nowise 
calculated  to  lead  into  error ;  it  did  not  assume  any  particular 
hypothesis  in  respect  to  universal  gravity,  or  attempt  to  declare 
its  laws  in  respect  to  distance,  but  left  them  for  future  investiga- 
tion, which  long  ere  this,  should  have  been  both  rational  and 
conclusive. 

It  is  said  of  Newton,  that,  after  receiving  the  hint  from  the 
falling  apple,  setting  out  with  the  law  of  Kepler,  namely,  "  that 
the  squares  of  the  times  of  revolution  of  the  different  planets,  are 
proportioned  to  the  cubes  of  their  distances  from  the  sun,"  he 
found  by  calculation,  that  the  force  of  solar  attraction  or  gravity, 
decreases  proportionally  to  the  square  of  the  distance. 

But  I  propose  to  show  that  if  he  set  out  with  Kepler's  said 
law,  he  very  soon  lost  sight  of  it ;  as  he  did  also  of  the  whole 
spirit  and  meaning  of  Kepler's  olher  law,  which  teaches  that  the 
motion  of  a  planet  in  an  eccentric  orbit,  varies  as  the  distance 
of  the  planet  from  the  sun  varies ;  which  last  great  law  of  Kepler, 


INTRODUCTION.  9 

Newton  must  have  wholly  misunderstood  or  misapprehended ; 
else  the  error  aforesaid  could  not  have  been  committed ;  for  I 
shall  endeavor  to  prove,  that  neither  Sir  Isaac,  nor  any  other  per- 
son, ever  found  the  force  of  solar  gravity  to  decrease  propor- 
tionally to  the  square  of  the  distance.  For.  if  such  be  the  case, 
Kepler's  second  law  should  be  so  far  altered  from  the  manner  in 
which  he  declared  it,  as  to  assert  that  the  motion  of  a  system  of 
planets  must  vary  proportionally  to  each  other,  as  their  respective 
distances  vary  from  the  sun,  thereby  wholly  overthrowing  the 
great  law  of  Kepler,  with  which  Newton  set  out,  as  it  would 
cause  the  squares  of  the  periods  of  a  system  of  planets  to  be  as 
the  fourth  powers  of  their  mean  distances  from  the  sun, — as  may 
be  readily  ascertained  by  the  most  obtuse  intellect.  For  a  fact 
more  easily  ascertained  scarcely  exists,  than  that  the  mean  motions 
of  a  system  of  planets  are  inversely  as  the  square  roots  of  their 
mean  distances  from  the  sun,  which  is  the  only  possible  condition 
in  which  the  squares  of  the  periods  can  be  as  the  cubes  of  their 
mean  distances  from  the  sun. 

Another  fact  as  simple,  is,  that  if  the  mean  motions  of  a  sys- 
tem of  planets  were  inversely  as  their  mean  distances  from  the 
sun,  and  also,  if  the  motion  of  each  individual  planet  varies 
inversely  as  its  distance  varies  during  its  period  or  revolu- 
tion, then,  indeed,  the  force  of  solar  attraction  would  be  inversely 
as  the  square  of  the  distance ;  and  the  periods  would  be  as  the 
squares  of  their  mean  distances  from  the  sun  ;  and  upon  no 
other  condition  whatever  could  the  force  of  solar  attraction  be 
inversely  as  the  square  of  the  distance. 

It  is  also  said  by  Newton's  biographer,  that  having  discovered 
the  law  of  gravity  of  the  planets  toward  the  sun,  he  endeavored 
to  apply  it  to  the  moon,  by  comparing  the  deflection  of  the  moon 
from  a  tangent  to  its  orbit,  with  the  law  of  falling  bodies  near 
the  earth  ;  and  that,  after  many  years  of  doubt,  he  at  length 
found  that  the  attractive  power  of  the  earth  varied  inversely  as 
the  square  of  the  distance  from  its  centre  to  the  moon  ;  and  that 
after  two  years  more,  spent  in  penetrating  the  consequences  of 
this  discovery,  his  immortal  Principia  came  forth,  of  which  it  is 
said,  that  not  more  than  two  or  three  of  his  contemporaries  were 
capable  of  understanding  it,  and  that  more  than  fifty  years  elapsed 
before  the  great  physical  truth  which  it  contained,  (namely,  his 
law  of  gravity,)  was  thoroughly  understood  by  scientific  men 
generally.  Nor,  indeed,  was  this  accomplished,  until  a  recon- 
ciliation to  the  law  was  brought  about  by  Clairaut,  by  one  of 
the  strangest  devices  of  which  ingenuity,  grown  desperate,  is  ca- 
pable, and  which  has  produced  a  kind  of  Gordian  knot  in  the 
2 


10  INTRODUCTION. 

science  of  astronomy,  which  we  had  much  better  cut  than  to 
pother  much  in  attempting  to  untie  it. 

But  perhaps  a  better  commentary  can  scarcely  be  made  upon 
our  inclination  to  adore  something  which  we  know  not,  than  that 
contained  in  a  prospectus,  issued  in  the  city  of  New  York, — by  a 
respectable  publisher,  this  year,  (1846)  for  publishing  in  this 
country,  for  the  first  time,  Newton's  Principia,  —  in  which  the 
publisher  remarks  as  follows  : 

"  It  is  a  singular  fact  that  this  remarkable  work,  —  the  Principia 
of  Sir  Isaac  Newton, — the  foundation,  at  once,  of  his  world- 
wide fame,  and  the  grandest  monument  of  human  intellect,  — 
should  have  remained  comparatively  unknown  in  this  country 
to  the  present  day.  Singular,  because  the  name  of  the  immor- 
tal author  is  familiar  as  a  household  word  ;  —  himself  honored, 
revered,  looked  up  to  as  one  of  the  demi-gods  of  by-gone 
times,"  &c.  "  But  the  Principia  has  hitherto  been  inaccessible  to 
popular,  use.  A  few  copies  in  Latin,  and  occasionally  one  in 
English,  were  and  are  to  be  found  in  some  of  our  large  libra- 
ries, and  in  the  possession  of  some  loving  disciple  of  the  great 
master.  A  dead  language,  however,  in  the  one  case,  and  an 
enormous  price  in  both,  particularly  in  the  ease  of  the  English 
edition,  have  thus  far  opposed  very  sufficient  obstacles  to  any 
general  circulation  of  the  work,"  &c. 

And  such  is  the  blind  idolatry  of  a  world,  thus  worshipping 
a  fellow-mortal,  which  it  may,  at  some  future  time,  wish  to  make 
the  scape-goat  of  its  errors. 

But  of  the  law  of  gravity,  I  propose  to  treat  in  its  proper  place ;. 
remarking  here,  however,  that,  notwithstanding  Sir  Isaac  New- 
ton, Bernoulli,  and  others,  have  strangely  endeavored  to  equate 
and  adjust  the  respective  attractive  powers  of  the  sun  and  moon, 
by  calculating  their  respective  influences  in  producing  the  tides, — 
in  which  Newton  estimates  the  influence  of  the  moon  to  that  of 
the  sun  as  3.5  to  1 ;  Bernoulli  as  2.5  to  1,  and  M.  Thouroud 
far  less  than  either,  —  nevertheless,  the  earth  revolves  farther  in 
its  orbit  in  one  day,  about  the  sun,  than  the  moon  does  about 
the  earth,  in  an  entire  revolution ;  which  circumstance  ought  to 
be  accounted  for  in  some  rational  and  proper  manner. 

But  Sir  Isaac  Newton  having  determined  his  law  of  gravity, 
physical  consequents  must  necessarily  follow  or  flow  from  it ;  as 
in  the  case  of  the  motion  of  the  moon's  apogee,  for  instance ;  and 
although  Sir  Isaac,  by  his  law  of  gravity,  had  furnished  but  just 
one  fourth  of  the  requisite  quantity,  or  only  sufficient  to  produce 
just  one  half  of  the  actual  or  observed  motion,  yet -Clairaut,  at 
length,  found  out  a  way  to  make  that  answer. 

So  also  in  respect  to  the  precession  of  the  equinoxes,  it  is  still 


INTRODUCTION.  11 

insisted  that  Sir  Isaac  Newton  assigned  the  true  physical  cause, 
namely :  the  attraction  of  the  sun  upon  the  surplus  matter  of 
the  earth,  over  and  above  its  greatest  inscribed  sphere ;  although 
Mr.  Vince  says,  "  it  is  acknowledged  that  Newton  fell  into  an 
error  in  his  investigations  as  to  the  amount  of  the  effect."  But 
this  acknowledgment  seems  rather  to  have  arisen  from  the  ne- 
cessity that  the  celebrated  astronomer,  Dr.  Bradley,  was  under,  of 
having  a  place  to  stand  on,  in  respect  to  his  discovery  of  the  nuta- 
tion of  the  poles  of  the  earth.  And  in  respect  to  the  theory  of  the 
tides,  I  think  Sir  Isaac  Newton  has  lost  nothing  from  La  Place's 
modification  of  the  cause  and  effect,  as  he  has  left  it  still  more  ab- 
surd and  inexplicable,  if  .possible,  than  he  found  it. 

As  my  principal  purpose  is  to  treat  of,  or  inquire  into  the 
nature  and  effect  of  the  more  direct  physical  causes  by  which 
the  worlds  are  more  immediately  governed  and  controlled  in  their 
eternal  rounds,  and  as  Sir  Isaac  Newton's  theory  of  light  does 
not  so  essentially  interfere  with  or  affect  those  laws,  I  shall  omit 
to  treat  of  that  department  in  his  philosophy,  as  that  may  be 
better  borne  with,  however  absurd  it  may  be,  than  his  theory  and 
law  of  gravity,  till  a  better  philosophy  shall  correct  the  absurdi- 
ties. But  not  so  in  respect  to  his  theory  of  universal  gravity,  (as 
by  him  taught  to  a  too  confiding  world,)  which,  like  some  nox- 
ious plant  or  vine,  has  interwoven  itself  into  the  whole  fabric  of 
astronomy,  disjoining  its  parts,  hiding  its  beauties,  and  affecting 
the  operations  of  all  its  physical  laws. 

This  is  the  theory  which,  while  it  asserts  that  all  gross  or  tan- 
gible matter  in  the  universe,  is  inert,  dead,  and  regardless,  asserts 
also,  that  all  matter  is  equally  endued  with  an  innate  or  inherent 
principle  for  the  attraction  of  other  matter ;  and  hence,  that  inert 
matter  is  made  to  possess  innate  powers  and  innate  tendencies. 
This  is  the  theory  which  requires  the  heavenly  bodies  to  be  pro- 
jected in  the  dkection  of  a  tangent  to  the  orbit  in  which  they 
are  to  revolve;  —  not  that  the  planet  is  to  pass  in  the  direction 
in  which  it  is  projected ;  the  projection  being  only  one  of  the 
resulting  forces,  the  other  being  inherent  in  inert  matter,  con- 
stantly operating,  though  at  a  vast  distance,  and  that  also 
through  space,  and  which  must  forever  after,  take  the  whole 
charge  of  the  body  so  projected,  and  safely  guid-e  it  through  all 
the  disturbing  forces,  during  its  eternal  rounds.  This  is  the  the- 
ory which  has  converted  the  whole  universe  into  a  system  of 
disturbing  forces,  and  the  whole  corps  of  astronomers  into  an 
exploring  party,  whose  business  it  is  to  search  out  and  regulate 
disturbances.  And  it  would  necessarily  be  so,  if  the  theory  were 
true;  for  the  sun  would  then  so  disturb  the  original  projectile 
force,  that  the  planet  could  not  pass-  at  all  in  the  line  of  projec- 


12  INTRODUCTION. 

tion,  and  the  planets,  in  turn,  would  disturb  their  neighbors,  to- 
gether with  every  stranger  that  came  into  their  vicinity.  Never- 
theless, the  disturbances  are  mutual,  and  not  at  all  prevented  by 
the  inertia  of  matter. 

But,  to  me,  this  seems  quite  too  fortuitous,  if  not  absolutely 
atheistical ;  and,  therefore,  I  must  leave  it  to  others  to  suppose 
that  a  universe,  governed  and  controlled  by  no  other  power  than 
that  arising  from  disturbing  forces,  would  operate  better  than  one 
governed  and  controlled  by  immutable  laws  of  order  and  regu- 
larity. Nevertheless,  I  do  not  accuse  Sir  Isaac  Newton  of  en- 
tertaining atheistical  views  or  notions,  for  he  even  wrote  upon  the 
prophecies  of  Daniel,  and  upon  the  Apocalypse,  and  were  he 
now  living,  would  probably  be  the  best  expounder  extant,  of 
their  mysteries. 

And  finally,  it  is  this  theory,  which,  from  the  time  of  its  pro- 
mulgation, has  served  (with  the  world)  "  to  make  the  worse 
appear  the  better  reason,  to  perplex  and  dash  maturest  counsels," 
whereby  astronomers  have  appeared  to  lose  sight  of  first  prin- 
ciples, and  rather  implicitly  to  follow  the  persuasive  sugges- 
tions of  a  file  leader,  with  a  zeal  that  would  have  been  far  better 
employed  in  the  ways  of  truth,  and  in  its  development  to  the 
world. 

And  now,  if  it  should  be  thought  that  I  have  been  somewhat 
severe  in  my  remarks,  it  ought  also  to  be  understood  that  I  have 
much  idolatry  to  contend  with,  and  even  an  apparent  determina- 
tion neither  to  behold  nor  acknowledge  any  error  in  anything 
Sir  Isaac  Newton  may  have  presumed  to  teach,  promulgate  or 
find  ;  and  hence  we  so  often  find  the  naked  allegation, — even  in 
the  highest  authorities,  as  if  it  were  "  confirmation  strong  as 
proofs  of  holy  writ," — that  Sir  Isaac  Newton  found,  &c. 

But  perhaps  what  I  have  said  may  only  serve  to  show  that  my 
faith  in  the  infallibility  of  Sir  Isaac  Newton,  is  not  so  great,  as 
that  of  some  others ;  and  if  a  further  expression  of  such  want 
of  faith  should  be  required,  I  will  here  declare,  that  I  have  yet 
to  learn  of  a  single  beneficial  discovery  or  principle,  origin- 
ally promulgated  by  Sir  Isaac  Newton,  in  anywise  beneficial 
to  physical  astronomy. 

But  I  will  not  attempt  to  justify  raillery,  or  irreverence  upon 
the  ground  that  others  before  me  have  used  such  weapons. 
Nevertheless,  Sir  Richard  Phillips,  (who,  perhaps,  has  made 
rather  a  feeble  attack  upon  the  stronghold,  and  who,  perhaps, 
should  have  adduced  more  efficient  evidence  to  justify  his  prov- 
ocation,) in  speaking  of  the  Newtonian  method  of  deducing  the 
law  of  gravity  from  a  consideration  of  the  deflection  of  the  moon 
from  a  tangent  to  its  orbit,  speaks  in  this  wise : 

"  How  gross  the  imposture,  then,  of  founding  a  system  on  a 


INTRODUCTION.  13 

fictitious  fall  of  sixteen  feet  per  minute  !  "  &c. — "  How  juggling, 
too,  to  compare  a  paper  fall  at  the  moon,  with  a  real  fall  at  the 
earlh ;  and  how  delusive,  even  under  that  juggle,  to  take  a  minute 
when  even  the  juggle  itself  would  hold  for  no  other  time ! "  &c. 
"  But  the  wonder,  after  all,  is  far  less,  that  a  system-maker  should 
be  seduced  to  make  the  assertion,  than  that  mathematicians,  and 
men  of  moral  worth  should,  from  1687  to  1831,  have  been  mis- 
taken enough  to  believe  and  teach  such  nonsense  with  solemn 
emphasis." 

The  fact  is,  that  Sir  Richard,  and  also  thousands  of  others, 
perceived  that  there  was  discrepancy  and  error  in  the  Newtonian 
promulgation  of  the  law  of  gravity;  and  strange  indeed  have 
been  the  devices  and  attempts  to  correct  or  set  aside  the  error ; 
as  also  to  reconcile  the  law  with  observed  phenomena,  in  lieu  of 
reinvestigating  the  Newtonian  method,  and  thereby,  at  once, 
detecting  the  error  which  Sir  Isaac  committed. 

Bat  to  me,  Sir  Isaac  Newton's  thoughts  often  appear  to  have 
been  extremely  crude  and  unanalyzed,  and  in  them  he  often 
forgot  his  own  philosophy,  such  as  it  was  ;  as,  for  instance,  when 
he  wished  to  be  wiser  than  Galileo,  in  respect  to  the  direction  a 
body  would  take  in  its  fall  from  a  state  of  rest  towards  the  earth, 
—  he  states  that  in  some  few  hundred  feet  fall,  it  would  advance 
some  half  inch  eastward,  in  consequence  of  the  diurnal  motion  of 
the  earth  ;  and  some  modern  experimenters,  (believing  such  to  be 
the  law,)  have  conceived  that  they  have  detected  the  half  inch 
advance  ;  but  a  more  whimsical  error  could  not  have  been  ad- 
vanced, if  we  suppose  the  law  of  gravity  to  be  an  omnipresent 
law.  But  we  are  too  ready  to  transgress  against  reason,  if  we 
suppose  the  law  to  be  with  us. 

Unity  of  purpose  in  the  Divine  Architect  of  the  universe,  is  the 
hypothesis  in  which  I  would  involve  all  philosophy  in  respect  to 
cause  and  effect ;  nor  would  I  be  too  hasty  in  adopting  conclu- 
sions which  might  savor  of  fortuity,  instead  of  being  the  result 
of  general  and  immutable  laws.  And  whether  some  philoso- 
phers of  high  repute,  since  the  days  of  Copernicus,  Galileo,  and 
Kepler,  have  sufficiently  discarded  fortuity  or  chance  from  being 
co-workers  in  the  operations  of  nature,  is  a  subject  worthy  of 
consideration  ;  and  if  not,  it  would  not  be  a  wonder  if,  thereby, 
the  universe  had  been  thrown  a  little  out  of  gear ;  and  if  it 
should,  in  consequence,  require  much  equating  to  enable  it  to 
fulfil  what  would  seem  to  be  its  proper  operations  or  destination. 

The  Copernican  system  of  astronomy  was  not  only  the  result 
of  great  sagacity,  at  a  time  when  much  ignorance  prevailed 
upon  the  science  of  astronomy ;  but,  also,  of  a  boldness  border- 
ing on  temerity,  in  venturing  to  declare  this  theory  to  a  world 


14  INTRODUCTION. 

full  of  superstition  and  bigotry,  and  wedded,  as  the  world  al- 
ways is,  to  those  notions  which  have  long  been  cherished  by  the 
multitude  as  matters  of  implicit  faith. 

Galileo  did  excellent  service  to  the  science  of  astronomy,  not 
only  in  his  discoveries  by  aid  of  the  telescope,  but  in  his  other 
philosophical  researches;  and  especially,  in  ascertaining  and 
determining  the  law  of  falling  bodies,  near  the  earth. 

But  Kepler  seems  to  have  been  the  first  who  entertained 
rational  notions  of  physical  astronomy ;  at  least,  the  first  who 
obtained  any  knowledge  of  those  immutable  laws,  based  upon 
or  founded  in  the  immutability  of  numbers,  by  which  the  solar 
system  is  governed  in  its  harmonious  movements ;  which  laws, 
so  by  him  discovered,  from  their  great  utility  to  the  science  of 
astronomy,  are,  in  great  justice  to  their  discoverer,  called  Kepler's 
laws.  And  happy  would  the  circumstance  have  been  for  the 
science  of  astronomy,  had  those  who  succeeded  those  master- 
builders,  and  essayed  to  erect  the  superstructure,  been  careful  to 
square  their  work  by  the  foundation  so  skillfully  and  admirably 
laid. 

Those  master-builders  did  not  assume  to  build  upon  hypo- 
thesis alone,  but  upon  well-ascertained  and  demonstrated  facts. 
Thus  Kepler, —  in  announcing  to  the  world  the  law  of  gravity, — 
did  not  attempt  to  announce  a  specific  law,  by  which  matter 
operates  upon  matter,  by  means  of  an  equal,  innate,  or  inher- 
ent gravity  ;  nor  did  he  attempt  to  promulgate  a  law,  by 
which  matter  operates  upon  matter,  through  distance  or  space ; 
and  hence  his  announcement  was  not  calculated  to  mislead 
mankind  in  respect  to  a  naked  hypothesis,  nor  to  cause  them 
to  err  by  an  erroneous,  pretended  demonstration,  either  of  which, 
to  be  sustained,  would  require  the  laws  of  nature  to  give  way, 
or  at  least  to  subscribe  liberally  to  their  support,  to  the  neg- 
lect of  those  matters  more  directly  submitted  to  their  charge  and 
control. 

But  in  respect  to  Newton's  broad  hypothesis, —  a*s  to  the 
equal  innate  gravity  of  all  matter,— may  it  not  too  strongly 
imply  a  necessity,  in  matter  itself,  to  possess  certain  innate, 
inherent,  or  independent  qualities,  sufficient  fo?  its  own  govern- 
ment ?  Besides,  if  the  law  of  force  be  an  inherent  and  inde- 
pendent quality  or  principle  of  matter,  it  may  be  hard  to  under- 
stand why  motion  should  not  also  be  an  inherent  quality  or 
principle  of  matter.  But  if  the  Newtonian  hypothesis  be  not 
true,  then  what  has  been  supposed  to  be  different  densities  of 
the  planets  may  be  accounted  for  upon  the  principle,  that  uni- 
form laws  operate  upon  matter  alike,  under  like  circumstances  ; 
which,  manifestly,  could  not  be  the  case  upon  the  Newtonian 


INTRODUCTION.  15 

% 

theory,  —  by  which  the  worlds  were  launched  on  their  never* 
ending  way  by  one  original  impulse,  thence  to  be  left  wholly 
dependent  upon  their  equal  inherent  gravity  for  their  future  regu- 
lation and  control,  notwithstanding  the  infinity  of  disturbing 
forces  forever  operating  from  this  supposed  power  of  innate 
gravity. 

But  what  is  certainly  of  vast  importance  to  the  science  of 
astronomy  is,  whether  the  force  of  gravity,  which  retains  the 
planets  in  their  orbits,  operates  inversely  as  the  distance,  or  in- 
versely as  the  square  of  the  distance,  or  in  some  other  ratio. 

It  is  quite  certain  that,  attempting  to  bring  the  Newtonian 
hypothesis,  of  the  force  of  attraction  or  gravitation,  to  agree  or 
correspond  with  facts  obtained  from  observation,  has  cost  the 
learned  much  trouble  in  exploring  for  equations,  by  which  to 
reconcile  and  harmonize  the  small  deviations  from  the  law,  as 
prescribed  or  promulgated. 

To  be  sure  Clairaut,  Euler  and  others  protested  for  a  long 
time  against  the  law,  as  promulgated  by  Newton  ;  for  that  the 
cause  would  not  be  adequate  to  the  effects  produced  by  the 
law,  —  as  in  case  of  the  motion  of  the  moon's  apogee ;  and, 
for  a  time,  the  moon  seemed  well  nigh  divorced  from  all  obliga- 
tion to  this  earth,  until  Clairaut,  — by  a  device  so  fraught  with 
ingenuity  that  few,  perhaps,  will  volunteer  to  examine  its  truth 
or  error,  —  declared  that  he  had  reconciled  the  Newtonian  law 
of  gravity  with  the  motions  of  the  moon.  This  seemed  to  be 
the  last  great  struggle.  Opposition,  in  all  quarters,  appeared  to 
give  way,  and  philosophy  seemed  not  only  compelled,  but 
rather  inclined,  to  rest  from  opposition  and  strife,  and  to  content 
itself  with  exploring  for  disturbing  forces  and  equations,  by 
which  the  Newtonian  law  of  gravity  might  be  justified,  however 
the  harmony  of  nature  might  be  torn  in  pieces  to  effect  the  pur- 
pose. And  notwithstanding  many  since  have  felt  in  their 
minds  to  rebel  against  so  unjust  and  arbitrary  a  law,  —  as  in 
case  of  Lord  Monboddo  and  Sir  Richard  Phillips,  as  well  as 
thousands  of  students  who  have  been  disciplined  under  this 
law,  —  yet  no  effectual  force  has  ever  been  organized  to  resist 
it.  While,  on  the  contrary,  the  superstructure  has,  for  near  two 
centuries,  been  erecting  upon  the  Newtonian  foundation  ;  in 
which  the  builders  have,  figuratively  'at  least,  made  use  of 
the  calculus  for  stones,  and  fluxions  for  mortar,  until  their  lan- 
guage, either  spoken  or  written,  has  become  so  confounded, 
that  the  world  at  large  have  long  since  ceased  to  have  any  un- 
derstanding of  it  whatever;  while  the  builders,  (to  use  a  figure 
of  M.  Bailly,)  like  vultures  over  a  battle-field,  soar  one  above 
another  by  such  awful  flights,  that  one  can  scarce  discern  anoth- 


16  INTRODUCTION. 

er ;  while  mortals,  who  tread  the  earth  beneath,  can  scarcely  dis- 
cern the  lowest  of  them. 

But,  instead  of  contending  against  the  sufficiency  of  the  law 
of  gravity,  as  given  by  Newton,  or  endeavoring  to  reconcile  the 
law  thus  given,  with  facts  which  utterly  deny  its  truth,  had 
Clairaut,  Euler,  and  others,  taken  the  trouble  to  reexamine  the 
evidence  from  which  Newton  deduced  his  supposed  demonstra- 
tion, they  would  readily  have  discovered  where  and  how  he 
committed  his  error,  —  namely,  in  assigning  the  square  of  the 
distance  to  the  rate  of  force,  in  lieu  of  assigning  or  referring  it 
to  the  amount  of  the  fall  of  the  body,  in  accordance  with  the  law 
of  falling  bodies  near  the  surface  of  the  earth.  This  one  cor- 
rection in  the  premises  might  have  led  to  a  further  and  better 
consideration  of  the  law  of  force  or  gravity,  and  eventually  to  a 
true  development  of  its  principles,  as  applied  to  the  control  of  a 
revolving  body ;  by  which  they  would  have  discovered  that  the 
moon  is  entitled  to  just  four  times  the  force  allowed  to  it  by 
Newton's  law  of  gravity,  and,  consequently,  to  twice  the  motion 
which  his  force  would  give. 

But  another  great  error  in  Newton's  calculations  (and  which 
will  be  considered  in  its  proper  place)  was,  in  supposing  that 
the  deflection  of  the  moon  from  a  tangent  to  its  orbit,  in  a  given 
time,  would  be  just  equal  to  the  fall  of  a  body  from  a  state  of 
rest,  at  the  distance  of  the  moon,  in  the  same  time. 

Neither  the  limits  nor  design  of  this  work  permit  me,  were  I 
disposed,  to  enter  the  lists  with  philosophers,  (either  ancient  or 
modern,)  who  have  theorized  upon  the  phenomena  of  motion, 
with  much  curious  speculation,  even  to  the  doubting  of  its 
existence.  For  the  same  reason  I  shall  not  go  into  the  inquiry, 
whether,  according  to  Newton's  theory,  matter,  by  nature,  is  inert, 
and  will  not  move  until  urged  by  some  adequate  force  ;  and 
that,  from  the  same  cause, —  inertia, —  when  once  put  in  motion, 
it  would,  if  not  counteracted  or  disturbed  by  some  other  force, 
forever  continue  to  move  in  a  right  line,  with  a  uniform  velocity. 

But,  perhaps,  after  all,  there  may  be  something  too  imaginary 
in  such  a  theory,  to  justify  its  being  considered  a  self-evident 
proposition.  For  on  recurring  to  the  laws  of  nature,  so  far  as 
we  are  able  to  investigate  them,  we  find  them  so  contrived  as 
often  to  forbid  what,  —  'to  our  fancies  or  imaginations,  —  might 
occur,  if  the  laws  of  nature  were  otherwise  than  they  are.  We 
know  of  no  such  thing  in  nature  as  that  of  a  body  moving, 
either  by  a  uniform  motion  or  in  a  straight  line  ;  for  all  matter 
not  only  moves  in  curves,  but  with  constantly  varying  veloc- 
ities. Nor  is  the  dilemma  to  be  avoided,  by  reference  to 
a  body  falling  toward  the  centre  of  the  earth  ;  for  the  motion 


INTRODUCTION,  17 

is  not  only  accelerated,  but  the  curve  which  it  represents  in  its 
fall  has  even  perplexed  La  Place,  and  other  great  mathemati- 
cians, to  describe  it.  Hence  it  might  seem,  that  those  who  have 
assumed  as  a  lemma,  or  self-evident  proposition,  that  a  uniform 
motion,  in  a  straight  line,  is  the  natural  state  or  condition  of  the 
motion  of  matter,  have  permitted  their  imaginations  to  devise 
quite  a  different  method  from  that  which  the  laws  of  nature  any 
where  permit. 

Similar  to  the  foregoing,  is  the  case  of  those  who  fancy  that 
the  natural  state  of  matter  is  a  state  of  rest ;  and  allege,  that  of 
the  truth  of  this  proposition,  also,  we  have  abundant  proof.  We 
know  of  no  matter  that  is  not  constantly  in  motion,  and  that, 
too,  in  accordance  with  what  appears  to  us  to  be  eternal  and 
immutable  laws.  To  be  sure,  a  stone,  or  other  ponderous  body, 
lying  upon  the  surface  of  the  earth,  appears  to  us  to  be  at  rest; 
but  who  does  not  know  that  it  is  the  tendency  of  such  a  body  to 
be  in  rr  otion  that  causes  it  to  remain  in  such  apparent  state  of 
rest,  r.nd  that  its  gravity  is  as  much  exerted  upon  the  body 
where  it  lies,  as  if  it  were  actually  falling  towards  the  earth  ? 

So,  also,  the  assumption  appears  to  have  been  gratuitous  and 
wholly  uncalled  for,  that  the  heavenly  bodies  must  necessarily 
have  been  projected,  as  with  a  view  to  motion,  in  straight  lines ; 
but  that,  by  the  power  possessed  by  inert  matter,  they  were 
wholly  prevented  from  moving  in  the  direction  in  which  they 
were  projected ;  and  this  upon  the  assumed  principle,  that  the 
forces  which  produce  motion  in  a  curve  are  resulting  forces,  — 
that  is,  that  the  motion  in  a  curve  is  the  result  of  two  forces 
momentarily  applied ;  or  that  the  straight  line  which  the  body 
would  pursue,  were  it  not  deflected  from  the  same,  arises  from 
the  inertia  of  the  body  in  a  state  of  motion,  being  the  result  of 
force  long  since  expended ;  and  that  the  deflecting  force  is  con- 
stantly or  momentarily  operating  upon  the  inertia  of  the  body, 
as  though  it  were  in  a  state  of  rest.  But,  upon  the  principle  of 
resulting  forces,  it  may  be  hard  to  conceive  why  the  motion 
should  not  be  continually  or  momentarily  increased ;  especially, 
if  the  circle  be  what  many  have  conceived  it  to  be,  namely,  a 
polygon  consisting  of  an  infinite  number  of  sides  ;  in  which 
case,  the  deflecting  force  would  be  exerted  by  many  different 
impulses,  and,  consequently,  the  rate  of  motion  would  be 
constantly  changing.  Nevertheless,  the  motion  in  a  circle 
remains  as  uniform  as  in  a  straight  line ;  and  Lord  Monboddo 
says,  that  in  the  days  of  Aristotle  the  line  of  a  circle  was  con- 
sidered as  simple  as  that  of  a  straight  line  ;  and  in  accordance 
with  this  idea,  it  is  certain,  that  the  only  rational  conclusion  to 
be  deduced  from  the  attempts  of  those  who  would  ascertain  the 
3 


18  INTRODUCTION. 

periphery  of  a  circle,  by  a  redaction  of  its  inscribed  and  circum- 
scribed polygons,  is,  that  the  angles  of  the  polygons,  by  becom- 
ing more  and  more  obtuse,  at  length  wholly  vanish  in  the 
line  of  the  circle  ;  and,  hence,  that  where  a  tangent  touches  the 
periphery  of  the  circle,  it  forms  no  angle  whatever  with  the  line 
of  the  circle. 

But  we  nowhere  discover,  in  the  mechanism  of  the  heavens, 
that  force  is  exerted  by  impulses ;  and  it  would  seem  that  the 
composition  and  resolution  of  forces,  —  as  taught  upon  the  par- 
allelogram or  otherwise,  —  instead  of  teaching  or  inducing  such 
notions  of  force  and  motion,  as  being  applicable  to  the  heavenly 
bodies,  would  have  simply  taught  that  motion,  produced  by 
impulses,  is  as  the  square  root  of  the  force  applied.  But  how  the 
eternal  and  immutable  laws  of  nature  would  have  been,  had 
they  been  otherwise  than  they  are,  I  know  not ;  nor  is  my  imag- 
ination boundless  enough  to  comprehend  things  so  vast,  and  give 
the  information  to  mankind ;  and  should  I  attempt  it,  I  should, 
probably,  find  myself  in  "  endless  mazes  lost." 

In  reference  to  the  science  of  astronomy, — which,  from  its 
nature,  requires,  perhaps  more  than  any  other  science,  the  aid  of 
mathematics,  or  rather  the  application  of  numbers  in  the  de- 
velopment of  its  truths,  —  may  it  not  be  questioned,  whether 
a  greater  regard  should  not  have  been  had  to  the  edification  of 
those,  whose  lack  of  leisure  and  of  other  facilities,  too  often  de- 
prive them  of  the  requisite  advantages  for  investigating  the  more 
abstruse  sciences,  and  which  might  be  afforded  them,  were  the 
learned  inclined  to  stoop  a  little  more  to  their  necessities  in  that 
respect  ? 

Thus  the  mathematical  languages,  if  I  may  so  speak,  in 
which  the  science  of  astronomy  is  now  generally  taught,  are 
become  absolutely  terrific  to  the  great  mass  of  mankind  ;  by 
which  means  much  of  the  science  is  to  them  but  little  if  any 
better  than  Rosicrusian  mysteries  ;  which  fact  has  a  tendency  to 
cause  their  knowledge  of  the  science  to  become  less,  in  propor- 
tion as  the  science  is  supposed  to  become  advanced.  For  I  will 
ask,  what  proportion  of  a  community  can  be  expected  to  obtain 
a  knowledge  of  algebra,  fluxions,  or  the  calculus,  or  an  infinity 
of  abstruse  geometrical  diagrams,  with  the  view  of  obtaining 
a  knowledge  of  the  science  of  astronomy,  whereby  they  may 
become  able  to  distinguish  for  themselves  the  beauty  of  truth 
from  the  deformity  of  error  ? 

But  happy,  however,  may  it  have  been  for  some,  that  they 
were  thus  enabled  to  obscure  their  philosophy,  even  to  such  a 
degree  that  the  learned  themselves  were  unable  to  penetrate  or 
dispel  the  dark  fogs  and  mists  that  pervaded  it.  And  it  is 


INTRODUCTION. 


19 


almost  to  be  feared,  that  the  libraries  of  the  present  day  will 
share  the  fate  of  the  Alexandrian,  from  so  universal  an  ignorance 
of  their  contents,  or  from  our  not  being  able  to  understand  and 
appreciate  them. 

Neither  Galileo,  nor  Kepler,  knew  aught  of  the  differential 
or  the  integral  calculus,  or  of  fluxions  ;  nor  had  Des  Cartes 
applied  algebra  to  astronomical  investigations.  Kepler  made 
use  of  the  then  popular  method,  by  the  use  of  the  Arabic 
characters ;  and  that,  too,  even  without  the  assistance  of  log- 
arithms. But  since  the  days  of  Kepler,  I  know  not  of  a 
single  discovery  in  physical  astronomy  which  deserves  the 
name. 

May  it  not,  then,  be  possible,  that  less  rigid  or  thorough 
mathematical  investigations  have  obtained,  in  consequence  of  the 
adoption  of  other  methods  than  would  have  been  used,  if  the 
popular  or  Arabic  character  had  been  retained  in  all  mathemat- 
ical investigations  ?  for  of  the  use  of  it,  all  must  have  some  know- 
ledge who  have  obtained  the  rudiments  of  an  education. 

And  may  it  not  even  now  be  questioned,  whether  the  ordin- 
ary mathematics,  by  popular  notation,  is  not  as  susceptible 
of  demonstrating  the  most  abstruse  problems  or  propositions, 
and  of  investigating  as  rigidly,  as  clearly,  and  with  as  much 
scrutiny,  the  arcana  appertaining  to  mathematical  science,  as 
any  of  the  substitutes  extant?  This  method  is,  certainly,  the  only 
one  with  which  the  mass  of  mankind  will  probably  become 
acquainted  ;  and,  hence,  to  adopt  into  the  sciences  such  a  method 
to  the  extent  of  its  powers,  would  certainly  bring  science  vastly 
more  within  the  reach  or  understanding  of  the  mass  of  man- 
kind, than  it  is  at  present ;  and  vastly  more  would  be  gained  to 
science  by  the  general  diffusion  of  knowledge,  than  perhaps 
would  be  lost  by  a  total  abandonment  of  all  other  methods. 

These  being  my  views,  I  shall,  in  the  present  work,  con- 
fine myself,  as  much  as  possible,  to  the  common  or  popular 
method  in  mathematics:  whether  it  be  in  attempting  to  disclose 
new  principles  in  the  economy  or  philosophy  of  numbers ;  or  in 
applying  them  to  the  investigation  of  the  laws  of  nature,  or  of 
scientific  truth  ;  making  use  of  as  few  additional  symbols  or 
characters  for  the  expression  of  abstract  ideas,  (in  addition  to  the 
natural  numbers,  or  popular  notation,)  as  the  subject  will  admit. 

It  is  not,  perhaps,  yet  known  to  what  extent  the  economy  of 
numbers,  (even  by  the  common  or  popular  method,)  may  be 
carried,  in  their  development  of  the  human  mind  in  its  upward 
progress;  which,  from  being,  in  their  simplest  application,  per- 
haps, the  most  abstract  and  simple  of  all  our  ideas,  may  also  be 
extended  into  the  most  intricate,  abstruse  and  confused.  For 


20  INTRODUCTION. 

although  unity,  or  individuality,  is,  perhaps,  as  distinct  a  concep- 
tion as  the  mind  entertains,  yet,  in  proportion  as  unities  or  in- 
dividualities are  increased  or  multiplied,  they  become  more  and 
more  indistinct  or  incomprehensible  to  the  mind ;  so  that  few, 
perhaps,  have  very  clear  conceptions  of  a  million. 

In  the  economy  of  numbers,  as  in  other  things  contrived  by 
infinite  wisdom  for  the  benefit  of  man,  there  is  a  fitness  of  things 
fully  commensurate  with  the  human  understanding;  and  to  a 
knowledge  of  which,  to  the  utmost  of  human  capability,  he 
seems  to  be  invited  and  urged  by  every  consideration  worthy  of 
his  unbounded  capacity,  when  fully  developed. 

Although  it  is  not  my  purpose,  in  this  place,  to  treat  particu- 
larly of  the  economy  of  numbers,  it  may,  nevertheless,  not  be 
wholly  unprofitable  to  commence  with  it,  and  refer  to  some  of  the 
qualities  of  unity  or  1,  and  their  peculiar  application  to  the  in- 
vestigation of  the  immutable  laws  of  nature. 

Unity,  then,  or  number  1,  seems  wonderfully  calculated  or 
adapted,  as  a  numerical  standard,  for  all  quantities  capable  or 
susceptible  of  numerical  value  or  admeasurement ;  as  of  bulk 
or  magnitude,  weight,  space,  time,  force,  motion,  distance,  &c.; 
for,  however  great  or  small  the  quantity,  it  may  be  denoted  by 
unity,  which  is  at  the  same  time  susceptible  of  indefinite  math- 
ematical or  numerical  divisibility.  But  what  peculiarly  dis- 
tinguishes unity  from  all  other  numerical  quantities,  and  makes 
it  the  standard  numerical  quantity  by  which  to  compare  all 
others,  is,  that  its  powers  and  roots  neither  augment  nor  diminish 
it.  This,  together  with  the  constant  reciprocity  which  exists  in 
the  elements  of  the  heavenly  bodies,  as  connected  with  the  laws 
by  which  they  are  governed,  whether  it  be  in  reference  to  the 
elements  of  an  individual  planet  of  a  system,  or  of  all  the  several 
planets  of  a  system  when  compared  with  each  other,  makes  it 
of  the  first  importance,  in  facilitating  our  operations  and  in 
arriving  at  satisfactory  results,,  that  unity  be  adopted  as  the  stand- 
ard measure  of  all  numerical  quantities  or  values.  This  being  the 
case,  by  always  regarding  unity  as  the  radical  centre  of  all  nu- 
merical quantities,  the  laws  which  govern  the  planetary  world 
will  be  found  to  be  based  upon  much  more  uniform  and  simple 
principles  than  they  have  in  general  been  supposed  to  ber  all 
harmonizing  and  agreeing  with  the  Keplerian  law,  that  the 
squares  of  the  periods  are  as  the  cubes  of  the  mean  distances 
from  the  sun.  Thus,  in  any  system  of  bodies  revolving  about 
a  central  force,  as  in  case  of  the  primary  planets  about  the  sun,, 
or  of  satellites  about  a  primary  planet,  either  of  the  revolving, 
bodies  may  be  taken  at  the  mean  distance  of  unity  or  1,  from 
the  central  force,  by  which  to  compare  the  elements  of  all  the 


I 

INTRODUCTION.  21 

other  revolving  bodies,  —  their  respective  distances  being  pro- 
portioned to  that  the  distance  of  which  is  taken  at  unity.  And  in 
such  case,  each  and  every  element  of  that  planet  which  is  adopt- 
ed as  the  standard,  —  or  the  distance  of  which  is  assumed  at 
unity,  — ,  must  also  be  assumed  at  unity,  in  order  that  the  Kep- 
lerian  law  may  be  fulfilled. 

In  respect  to  whatever  of  theoretic  matter  I  may  introduce,  as 
supposed  deductions  and  conclusions  drawn  by  way  of  analogy 
from  known  facts  and  laws,  I  shall  feel  no  anxiety  in  sustaining 
the  same,  further  than  it  shall  be  sustained  by  legitimate  evi- 
dence ;  as  in  case  of  my  theory  in  respect  to  the  identity  between 
magnetism  and  the  attraction  of  gravitation  by  which  the  heav- 
enly bodies  are  controlled,  of  which  I  have  conceived  many 
evidences  to  exist  in  the  phenomena  of  nature.  In  support  of 
this  theory,  I  shall  adduce  the  evidence  arising  from  the  differ- 
ent powers  of  attraction  of  the  different  planets,  in  proportion  to 
their  respective  bulks  or  magnitudes  ;  and  in  this  case,  I  shall  at- 
tempt to  show  that  the  phenomena  arises  from  a  general  law, 
(not  fortuitous)  deducible  from,  the  identity  of  gravitation  with 
magnetic  attraction. 

I  shall  call  in  the  aid  of  the  phenomena  of  the  tides,  arising  from 
the  attractive  power  of  the  moon ;  and  shall,  on  this  principle, 
endeavor  to  account  for  many  of  the  phenomena  in  respect  to 
the  tides,  in  a  very  different  way  and  manner  from  those  com- 
monly adopted  respecting  them ;  and  show  why  equal  tides  can 
exist  on  opposite  sides  of  the  earth  at  the  same  time ;  why  the 
tide-waters  of  broad  oceans  pile  high  upon  the  shores  of  broad 
continents,  while  they  are  comparatively  small  on  the  shores  of 
the  narrow  parts  of  those  continents,  and  still  smaller  in  the 
midst  of  broad  oceans,  where  only  small  islands  exist,  &c.  ;  and 
will  endeavor  to  account,  from  the  same  simple  cause,  for  the 
reason  why  high  tide  is  not  as  far  behind  the  moon  in  the  midst 
of  broad  oceans,  as  upon  the  shores  of  broad  continents ;  why 
large  inland  seas  produce  little  or  no  perceptible  tides,  &c. 

In  support  of  the  identity  of  gravitation  and  magnetism,  to 
some  extent,  at  least,  I  propose  to  call  to  my  aid  the  nice  and 
critical  observations  of  Dr.  Bradley,  (Astronomer  Royal  to  Geo. 
II,)  made  upon  some  of  the  fixed  stars,  with  the  view  of  ascer- 
taining whether  they  had  any  sensible  parallax  ;  in  which 
he  finally  deduced  the  hypothesis  of  the  aberration  of  light, 
which,  in  astronomy,  has  become  technical  in  its  use  and  signi- 
fication, denoting  a  small  apparent  motion  of  ihe  fixed  stars,  oc- 
casioned by  the  progressive  motion  of  light  and  the  earth's  an- 
nual motion  in  its  orbit.  And  although  the  hypothesis  has, 
since  his  announcement  of  it,  been  elaborated  into  a  sys- 


22  INTRODUCTION. 

tern  or  branch  of  philosophy,  by  the  aid  and  assistance  of  the 
higher  branches  of  mathematics,  and  its  supposed  consequences 
or  effects  introduced  into  astronomical  observations  and  calcula- 
tions, I  must  declare  my  full  dissent  to  the  correctness  of  the 
conclusion  or  hypothesis,  believing  the  same  to  be  in  nowise 
justified  by  the  premises  from  which  it  is  drawn. 

In  these  introductory  remarks  I  will  refer  to  but  one  other 
subject  connected  with  my  theoretic  speculation,  as  collateral 
or  rather  presumptive  evidence  in  favor  of  that  unity  of  pur- 
pose, and  fitness  of  things,  which  all  will  readily  ascribe  to  the 
Divine  Architect  of  the  universe.  I  refer  to  the  subject  of  Geol- 
ogy, a  subject  which,  though  recent  as  a  science,  or  subject  of 
investigation,  abounds  with  that  multiplicity  of  facts  and  phe- 
nomena which  has  already  caused  theories  in  respect  to  it,  to 
succeed  each  other  in  rather  rapid  succession,  from  the  circum- 
stance of  a  few  newly  discovered  facts  overturning  the  last 
formed  theory. 

But  the  Huttonian,  or  Volcanic  theory, — that  the  surface  of  the 
earth  is  but  the  cooled  shell  of  what  was  once  an  entire  mass 
of  fused,  or  melted  matter,  and  that  all  the  supposed  catastro- 
phes that  have  happened  to  the  earth,  in  producing  the  appear- 
ances and  phenomena  on  and  near  its  surface,  as  also  the  increase 
of  the  temperature  of  heat  as  we  descend  into  the  earth,  &c.,  which 
are  conceived  to  be  the  evidences  of  such  hypothesis, — is,  in  the 
opinion  of  many,  so  deeply  laid  as  not  to  be  easily  overturned. 
But,  to  me,  this  hypothesis  seems  altogether  too  fortuitous,  both 
in  its  origin  and  consequences,  to  obtain  credence  for  a  moment, 
from  one  who  believes  creation  to  be  sustained  by  immutable 
and  uniform  laws;  and  seems  far  better  calculated  to  furnish 
food  for  the  marvellous,  than  to  lead  mankind  to  very  high  no- 
tions in  respect  to  the  wisdom,  power,  and  goodness  of  their 
Creator.  It  would  seem  also,  as  if  there  would  exist  no  necessity 
for  such  an  hypothesis,  which  so  illy  accounts  for  the  phenomena 
to  which  it  is  assigned,  if  due  regard  were  had,  or  proper  consider- 
ation given,  to  what  were,  until  of  late,  occult  laws  or  operations 
of  nature, —  but  which,  in  modern  times,  have  been  deduced 
and  drawn  forth  by  the  sagacity  of  man,  not  as  unprofitable 
knowledge,  but  such  as  ministers  to  his  safety,  his  support  and 
his  happiness.  I  refer  to  the  modus  operandi  of  the  law  of  elec- 
tricity, in  the  experiment  by  Dr.  Franklin  with  the  string  of  a 
kite  ;  to  galvanism,  in  the  experiment  by  Galvani  with  the  leg  of 
a  frog ;  and  to  electro-magnetism,  arid  the  experiments  by  Oer- 
sted, of  no  less  sagacious  intellect. 

Those  operations,  or  laws  of  nature,  together  with  that  of 
magnetism,  (and,  perhaps,  the  operation  of  light,)  are,  in  general, 


INTRODUCTION.  23 

considered  by  philosophers  so  far  identical,  as,  at  least,  to  be 
modifications  of  one  great  and  universal  law,  the  constant  effect 
of  whose  silent  operations,  are  more  apparent  under  the  form  of 
gravitation  or  attraction,  than  under  any  other ;  and  that  this 
universal  law,  in  all  its  various  modifications  and  operations, 
acts  as  a  messenger  of  Omnipresence,  as  well  in  the  government 
and  control  of  inanimate  and  unorganized  matter,  as  in  that 
which  is  organized  and  animate,  and,  therefore,  dependent  upon 
final  causes.  We  see  the  operation  of  this  law  or  principle, 
in  gravity  or  attraction,  in  controlling  the  heavenly  bodies  in 
their  motions.  We  see  its  operation  in  magnetism  or  mag- 
netic attraction,  as  denoted  by  a  common  magnet  or  loadstone  ; 
presenting  that  wonderful  phenomenon  exhibited  in  the  polarity 
of  a  common  compass  needle,  —  accompanied  by  the  paradoxi- 
cal fact  or  phenomenon,  that  what  is  termed  the  north  pole  of 
the  needle,  is  attracted  by  the  south,  and  repelled  by  the  north 
pole  of  another  needle  ;  which  paradoxical  fact  1  propose  to  no- 
tice hereafter.  Here  then  we  discover  (in  the  absence  of  any 
more  appropriate  word  by  which  to  explain  the  occult  opera- 
tion) the  magnetic  fluid  operating  in  different  directions,  namely, 
towards  the  opposite  poles  of  the  earth,  (except  in  case  of  the 
variation  or  declination  of  the  needle,)  and  this  upon  all  parts 
of  the  surface  of  the  earth  ;  while  at  the  poles  of  the  earth,  it 
can  have  but  one  direction,  as  denoted  by  the  dip  of  the  needle. 

If,  then,  we  find  this  all-pervading,  all-powerful,  but  occult 
law,  thus  operating  throughout  the  surface  of  the  earth,  (and 
analogy  would  indicate  the  same  in  respect  to  other  planets,)  if 
we  find  it  operating  in  the  movement  and  control  of  the  vast 
bodies  which  compose  the  solar  system ;  if  we  find  it,  under  other 
modifications  or  appearances,  as  those  of  electricity,  galvanism, 
or  electro-magnetism,  (when  subjected  to  the  control  of  man, 
and  made  to  subserve  his  purposes  and  curiosity,)  operating  by 
its  own  immutable  laws  upon  a  small  scale,  as  that  of  the 
electric,  the  galvanic  or  electro-magnetic  battery,  or  of  the  tele- 
graph,— is  it  to  be  supposed,  or  presumed,  that  those  operations 
are  only  incident  to  human  contrivance  ?  or,  rather,  on  the  con- 
trary, are  not  those  laws,  or  principles,  designed  for  similar  but 
vast  and  continued  operations  in  the  control  of  the  universe, 
which,  otherwise,  must  be  left  to  fortuity  or  chance  ?  For  it  is 
inconceivable,  how  an  immutable  law  can  be  subjected  to  the 
performance  of  operations  not  in  accordance  with  its  common 
sphere  of  action. 

I  would  ask,  then,  if  what  are  termed  the  positive  and  nega- 
tive fluids  (for  here  I  will  use  no  other  expression  to  denote 
them)  of  this  all  powerful  agent,  (under  whatever  modification 


24  INTRODUCTION. 

it  may  operate,)  meet  in  concentration  at  the  centre  of  the  earth, 
whether  it  would  not  far  better  account  for  all  the  phenomena 
upon  and  near  the  surface  of  the  earth,  than  does  the  present 
hypothesis  of  internal  fires,  as  also  for  an  infinite  variety  of 
other  phenomena,  which  the  present  hypothesis  in  nowise  ac- 
counts for,  —  as  that  of  the  independent  coal  formation,  for  ex- 
ample ? 

Besides,  upon  the  theory  which  I  propose,  there  would  be 
nothing  fortuitous ;  all  would  result  from  the  operation  of  con- 
tinued laws ;  there  would  be  no  apprehension  that  the  internal 
fires  would  become  extinct ;  or  that  the  crust  might  give  way, 
and  produce  catastrophes,  such  as  the  imagination  alone  can 
conceive. 

As  no  excuse  is  pleaded  for  certain  courses  which  I  see  prop- 
er to  take,  or  pursue,  in  the  work,  hence  none  need  be  granted  ; 
—  as,  for  instance,  in  case  of  the  investigation  of  the  law  of 
gravity,  from  a  consideration  of  the  law  of  falling  bodies; — in 
which  case  the  text  on  which  I  comment,  (generally  taken  from 
Maclaurin,  Vince  or  Rees,)  may  not  only  appear  to  some  to  be 
too  often  and  too  liberally  quoted,  but  the  commentaries  may  seem 
to  savor  too  much  of  repetition,  all  tending  to  prove  one  and  the 
same  thing,  namely,  that  the  force  of  gravity  varies  inversely 
as  the  distance  varies.  It  is  but  few  subjects,  however,  of  which 
I  treat,  and  those  are  fundamental  and  important;  and  the 
course  I  have  pursued,  I  believe,  has  ever  been  esteemed  laud- 
able, in  regard  to  important  truths.  They  should  be  kept  before 
the  people,  until  their  importance  shall  be  duly  considered  and 
appreciated  ;  they  should  even  be  taught  by  way  of  inculcation  : 
and  if  error  is  to  be  removed,  to  make  room  for  the  truth,  the 
task  is  doubled.  We  should  "  try  every  art,"  like  a  bird  that 
"  tries  to  tempt  her  new  fledged  offspring  to  the  skies ; "  nor  be 
satisfied  with  some  formal  intimation,  believing  the  world  to  be 
under  obligation  to  hold  or  regard  it  as  truth,  or  perish.  And 
although  we  may  be  inclined  to  adore  whatever  human  devices 
may  have  established,  nevertheless,  truth  is  too  catholic  in  its  na- 
ture to  be  established  by  convention. 

It  is  not  expected  that  the  human  race  will  be  born  prodigies, 
or  even  possessing  much  intuitive  knowledge  ;  nor  is  it  ration- 
ally to  be  supposed,  that  a  precocity  will  soon  attach  to  human 
nature,  whereby  infancy  shall  be  the  wisest  portion  of  our  lives ; 
and  whenever  such  has  been  the  case,  and  any  have  become  the 
oracles  at  which  we  might  knock,  or  to  whom  we  might  apply  for 
wisdom,  —  we  have  beheld  the  sad  reality,  to  use  tire  language  of 
Job,  that  "wisdom  has  died  with  them."  No  philosopher  has 
yet  been  sagacious  enough,  (or  has  yet  had  opportunity,)  to 


INTRODUCTION. 


25 


detect  and  seize  the  ethereal  spark  and  blow  it  into  a  flame ;  to 
lay  hold  of  the  evanescent  thread  in  the  tissue,  and  to  unravel 
and  draw  it  forth,  for  the  edification  and  benefit  of  man, — 
as  in  case  of  some  of  the  other  great  laws  of  nature,  long 
hidden  and  occult,  as  those  of  electricity,  galvanism,  electro- 
magnetism,  &c. 

Nevertheless,  those  prodigies  always  seem  inclined  to  devel- 
ope  or  disclose  their  numerical  results,  by  means  of  legitimate 
numbers  ;  and,  hence,  to  the  mass  of  mankind,  they  are,  in 
some  respects,  equally  as  beneficial  and  far  more  intelligible,  than 
many  who  have  conceived  their  own  craniums  to  be  the  very 
archives  of  wisdom. 

Since,  then,  mankind  must  plod  on,  —  not  only  in  the  acquiring 
of  knowledge  from  experience,  but  also  from  the  stock  which  has 
been  accumulated  as  the  general  store, — it  is  scarcely  a  mark  of 
wisdom,  or  philosophy,  for  those  who  have  been  placed  as  file 
leaders,  in  the  great  march  of  human  improvement,  (or  who 
have  been  regarded  as  such,)  to  lead  off  so  far  in  advance,  that 
their  position  can  only  be  distinguished  by  the  reflection  of  their 
names  from  the  sky  ;  and  they  hence  become  no  guide  whatever 
to  the  multitude. 

But  as  physical  astronomy  ought  to  have  its  foundation  based 
on  truth,  and  as  the  involution  and  evolution  of  all  the  strange 
curves  in  which  modern  astronomy  compels  the  planetary 
system  to  wander,  are  subjected,  for  the  accuracy  of  their  in- 
flexure,  to  the  curve  of  the  circle,  it  is  certainly  desirable  that 
the  circle,  in  order  that  it  may  be  a  true  standard,  should  be 
mathematically  round.  And  in  order  to  furnish  such  standard, 
I  shall  proceed  to  an  investigation  of  the  quadrature  of  the  cir- 
cle, prior  to  a  direct  investigation  of  the  laws  of  force  and  mo- 
tion. 

In  my  investigation  of  the  quadrature,  (as  in  all  of  my  other 
investigations,)  I  have  endeavored  to  render  the  process  as 
simple  and  easy  as  possible,  carefully  avoiding  any  method 
that  may  not  at  once  be  made  familiar  to  any  one  having  the 
rudiments  of  an  education.  Hence  I  have  adopted  the  letter  p, 
to  express  or  denote  generally  the  word  polygon  ;  and  this  from 
the  circumstance  that  a  figure  at  the  right  of  the  letter,  may 
readily  denote  the  number  of  sides  of  the  polygon,  whenever  it 
may  be  necessary  to  give  the  number  of  sides.  And  this,  to- 
gether with  a  few  denotations  of  polygonal  quantities,  expressed 
by  letters  of  the  alphabet,  with  one  or  two  signs  definitely  ex- 
plained, is  all  the  analytical  machinery  which  I  propose  to  adopt 
or  use.  For  I  have  yet  to  learn,  that  the  beauty  of  truth  depends 
upon  its  robes,  however  flaunting  or  party-colored  they  may  be. 
4 


26  INTRODUCTION. 

Hence,  it  does  not  require  a  volume  of  fluxions,  and  the  cal- 
culus, to  explain  the  quadrature  to  mankind,  or  give  them  ra- 
tional notions  of  the  curve  of  the  humble  circle ;  for  all  agree, 
that  those  methods  will  not  disclose  the  ratio  between  the  diam- 
eter and  circumference.  Archimedes,  in  his  sagacity,  sought 
out  the  true  method ;  but,  having  fallen  into  an  error  in  the  oper- 
ation, he  missed  the  mark,  and  thence  sought  to  detect  the  error 
to  the  day  of  his  death  ;  but,  having  failed,  the  error  has  yet 
remained,  and  has  at  length  been  extended  to  a  prodigious 
length  of  tail. 

But  had  Archimedes  adopted  unity  as  the  diameter  of  the 
circle,  and  geometrized  upon  lines  in  lieu  of  areas,  he  would  at 
once  have  discovered  wherein  his  error  lay.  And  hence,  in 
respect  to  the  circle,  we  have  a  notable  instance  of  the  tenacity 
of  mankind,  in  adhering  to  and  sustaining  error,  (however  wide 
from  the  truth,)  if  they  imagine  the  law  to  be  with  them  ;  as  in 
the  case,  also,  of  Dr.  Bradley's  theory  of  the  Progressive  Motion 
of  Light  in  Time,  —  which  theory  has  placed  the  sun  some 
eight  and  one-fourth  minutes  of  time  out  of  its  apparent  place, 
—  which  theory  has  been  elaborated  by  fluxions  and  the  calcu- 
lus, until  no  one  is  justified  in  believing  his  own  eyes  ;  when, 
perhaps,  the  error  of  the  hypothesis  may  be  readily  shown  by 
the  simplest  diagram,  and  also  that  the  Dr.  drew  conclusions 
directly  the  reverse  of  the  phenomena  from  which  he  deduced 
them. 

But  an  instance  of  this  kind,  not  less  remarkable,  perhaps, 
than  any  other,  is  that  in  respect  to  the  Newtonian  law  of  grav- 
ity, —  the  demonstration  of  which,  perhaps,  many  suppose  to  be 
a  result  obtained  through  the  medium  of  fluxions,  the  calculus, 
the  problem  of  the  three  bodies,  or  by  some  mystical  operation 
of  corpuscles  or  physical  atoms  upon  each  other,  &c.,  when 
even  the  Newtonian  method,  on  which  he  labored  for  years,  was 
so  extremely  simple,  that  any  school-boy  may  reexamine  it,  and 
detect  the  error  which  Newton  made.  And  such  is  the  ease 
with  which  all  physical  or  mathematical  truths  may  be  demon- 
strated, whilst  error  often  requires  a  strange  and  uncommon  dress 
to  hide  its  deformities.  But 

"  Patron  or  intercessor,  none  appeared." 

"  And  I  looked,  and  there  was  none  to  help  ;  and  I  wondered 
that  there  was  none  to  uphold." 


UNITY    OF    PURPOSE,    ETC. 


CHAPTER     I. 

On  the  Quadrature  of  the  Circle. 

SECTION    FIRST. 

To  find  a  rational  proportion  between  the  diameter  and  cir- 
cumference of  the  circle,  which  might  be  definitely  expressed  in 
some  given  power,  or  root,  of  numbers,  has  long  been  a  desid- 
eratum with  mankind,  as  we  learn  not  only  from  tradition  and 
the  history  of  geometry  and  mathematics,  but,  also,  from  every 
elementary  work  upon  geometry  and  mathematics,  in  which  we 
readily  discover  that  a  very  great  portion  of  all  the  geometric 
problems,  propositions  and  demonstrations,  have  had  in  view 
to  ascertain  the  true  ratio  between  the  diameter  and  circum- 
ference of  the  circle,  as  though  it  were  the  one  thing  needful  in 
science. 

Nor  has  this  longing  for  a  paramount  truth  been  without 
its  beneficial  effects  on  science,  as  thereby  rectilinear  geometry 
has  been  developed  to  a  vast  extent ;  but  its  failure  to  develope 
the  true  quadrature,  has  elicited  expressions  like  the  following, 
even  from  modest  authors,  namely :  "  If  there  existed  a  ra- 
tional proportion,  —  that  is,  a  proportion  to  be  expressed  in 
whole  numbers,  —  of  the  surface  of  a  circle  to  a  square  surface, 
there  would  be,  at  the  same  time,  a  rational  proportion  between 
the  diameter  and  the  circumference.  But,  from  geometrical 
reasons,  no  rational  proportion  of  the  diameter  to  the  circum- 
ference is  possible ;  it  can  be  expressed  only  by  approximation. 


28  ON     THE    QUADRATURE 

However,  the  proportion  thus  obtained  is  quite  as  correct  as  is 
necessary  for  any  purpose  in  the  applied  mathematics."  But, 
however  modest  the  above  allegations  are,  compared  with  those 
of  some  authors,  I  must  dissent,  not  only  from  what  the  author 
alleges  as  truths,  but  also  from  the  consequence  and  effect  which 
such  kind  of  teaching  is  calculated  to  have  on  mankind. 

Why  should  the  author  speak  of  a  ratio  expressed  in  whole 
numbers,  when  even  the  ratio  between  the  linear  measures  of 
the  square  cannot  be  so  expressed,  as  that  between  the  diagonal 
and  circumference,  but  must  be  expressed  in  the  second  power 
of  numbers  ?  And  even  though  the  ratio  of  the  diameter  of  the 
circle  to  its  circumference  cannot  be  definitely  expressed  short  of 
Ihe  third  power,  —  or  some  power  higher  than  the  second,  —  it 
will  be  no  less  a  definite  expression  of  the  true  ratio,  than 
though  it  could  be  expressed  in  whole  numbers  in  the  first  pow- 
er. Nor  was  the  author,  perhaps,  aware,  how  much  more  bene- 
ficial to  science  the  true  ratio  would  be,  though  it  were  only 
capable  of  being  definitely  expressed  in  some  higher  power  than 
the  first  or  second. 

Thus  the  expression  of  the  author  is  fallacious.  But  when 
the  author  alleges  that,  "  from  geometrical  reasons,  no  rational 
proportion  of  the  diameter  to  the  circumference  is  possible,"  I 
think  the  allegation  altogether  too  bold  to  aid  science  in  its  pro- 
gress ;  and  it  is  an  allegation  certainly  uncalled  for,  by  those  who 
are  in  search  of  truth.  It  is  sufficient  that  geometry,  in  the 
hands  of  those  who  have  furnished  it  with  its  reasons,  has  not 
been  able  to  disclose  to  mortals  the  true  ratio  between  the  diam- 
eter and  circumference  of  the  circle. 

But  the  same  author,  with  all  his  wonted  modesty,  has  gone 
much  further,  and  made  use  of  allegations  far  more  reprehensi- 
ble, when  he  alleges  that  the  ratio,  obtained  by  a  supposed  ap- 
proximation, is  quite  as  accurate  as  is  necessary  for  any  purpose 
in  the  applied  mathematics.  I  have  yet  to  learn  that  funda- 
mental errors  are  as  good  as  the  truth,  let  such  doctrine  come 
from  what  source  it  may  ;  and  I  predict  that  the  popular  ratio 
will,  sooner  or  later,  be  found  not  quite  as  accurate  as  is  neces- 
sary. But  the  supposition  is,  that  geometry  has  not  only  furnished 
reasons  why  no  definite  proportion,  or  ratio,  can  exist  between 
the  diameter  and  circumference  of  the  circle,  but  that  it  has  been 
able  to  direct  mankind  to  an  error  so  near  the  truth,  that  it  is 
equally  as  good,  or  beneficial,  as  truth  itself.  Thus  geometry 
itself,  so  full  of  rectitude  and  truth,  must  be  bent  and  warped 
from  its  legitimate  station,  and  made  subservient  to  human 
error. 


OF    THE    CIRCLE.  29 

Nevertheless,  the  Pythagorean  theorem,  or  47th  problem  of 
the  1st  Book  of  Euclid,  with  its  infinity  of  mathematical  ap- 
plications, (as  conclusive  to  the  mind,  as  is  the  geometric 
demonstration  from  which  they  are  deduced,)  yet  remains,  a 
boon  to  science ;  thus  for  example, — that  the  diagonal  and  one  side 
of  the  square,  may  have  a  definite  ratio  expressed  in  the  second 
power  of  numbers.  That,  if  any  given  surface  be  conceived  to 
be  enlarged  or  diminished,  (still  retaining  the  same  form,)  the 
area  or  surface  will  be  enlarged  or  diminished,  in  a  duplicate 
ratio  to  that  of  any  given  line  or  measure  of  such  area.  That 
when  the  diameter  of  the  circle  is  unity,  the  area  is  equal  to  one- 
fourth  of  the  periphery  ;  and  that  the  area  of  any  circumscribing 
polygon  of  such  circle  is  equal  to  one-fourth  of  the  perimeter  of 
such  polygon,  and  that  the  area  of  the  circumscribed  square  is 
equal  to  the  diameter  of  such  circle.  That,  if  the  diagonal  of  a 
square  be  2,  or  twice  unity,  the  area  of  such  square  is  equal  to 
the  diameter  of  its  circumscribed  circle.  That,  as  the  area  of 
the  circumscribed  square  of  a  circle  is  double  that  of  the  area  of 
the  inscribed  square  of  such  circle,  so  the  area  of  the  circum- 
scribed circle,  of  a  given  square,  is  double  the  area  of  the  in- 
scribed circle  of  the  same  square,  for  the  proportions  of  their 
respective  diameters  are  the  same ;  and  that  one-fourth  of  the 
perimeter  of  any  regular  polygon,  multiplied  by  the  diameter  of 
the  inscribed  circle  of  such  polygon,  gives  the  area  of  such  poly- 
gon ;  and  that  the  same  holds  in  respect  to  the  circle. 

These,  together  with  the  infinity  of  corollaries  and  conse- 
quents, which  flow  from  legitimate  geometric  demonstrations, 
are,  doubtless,  sufficient  to  enable  mankind  to  ascertain  the  true 
quadrature,  by  the  aid  of  right  reason. 

Thus,  a  very  great  part  of  our  labor,  on  this  portion  of  the  sub- 
ject, must  consist  in  bringing  together  and  comparing  demonstrat- 
ed principles,  and  mathematically  arranging  them,  in  such  man- 
ner as  to  enable  us  to  arrive  at  the  required  result.  This,  perhaps, 
may  be  done,  if  we  take  proper  heed  to  those  very  appropriate 
remarks  of  M.  Legendre,  in  which  he  alleges,  that  "  by  deducing 
consequences  from  one  or  more  propositions,  we  may  be  led 
back  to  some  proposition  already  proved  ;  and  that  an  indubita- 
ble proof  of  their  certainty  is,  that  however  we  combine  them 
together,  provided  only  our  reasoning  be  correct,  the  results  we 
obtain  are  always  perfectly  accurate."  Such  language  differs 
widely,  however,  from  that  actual  abomination  to  science,  to 
which  the  same  author  stoops,  when  he  alleges,  that iu  the  prob- 
lem of  the  quadrature  is  now  degraded  to  one  of  those  idle 
questions,  about  which  no  one  possessing  the  least  tincture  of 


30  ON    THE    QUADRATURE 

geometrical  science,  will  spend  any  portion  of  his  time ;  that 
although  the  true  ratio  of  the  diameter  to  the  circumference  has 
not  been  obtained,  yet,  that  no  advantage  whatever  would  be 
derived  from  the  true  ratio,  over  the  approximate  ratio,  &c. ; " 
thus  boldly  alleging  that  error  is  equally  beneficial  with  the  truth. 
And  I  will  here  ask  of  science,  (and  perhaps  the  disciples  of  Co- 
pernicus can  answer,)  if  any  known  distance  from  the  truth 
can  well  be  ascertained,  if  we  know  not  where  the  truth  lies  ? 

The  idea  or  conception  of  invoking  the  aid  of  geometry,  for 
the  purposes  of  a  true  induction  into  a  polygonal  progression, 
whereby  the  ratio  between  the  diameter  and  circumference  of 
the  circle  might  be  obtained,  is  of  high  antiquity,  as  such  was 
the  method  adopted  by  Archimedes.  And  it  is  generally  ac- 
knowledged, that  no  essential  improvement  has  yet  been  made 
either  in  his  process  or  result,  otherwise  than  by  means  of  certain 
devices  for  facilitating  what  is  supposed  to  be  a  true  approxima- 
tion towards  the  truth;  and  it  seems  to  have  been  taken  for 
granted,  that  Archimedes  was  satisfied  with  his  method,  and  the 
results  arising  therefrom.  But,  in  my  view,  it  is  wholly  unrea- 
sonable to  suppose  that  Archimedes,  after  having  written  a  treat- 
ise upon  the  quadrature,  if  he  had  been  satisfied  with  the  prin- 
ciples upon  which  he  had  proceeded,  or  with  the  results  arrived  at, 
would,  at  the  age  of  seventy-five  years,  have  been  poking  in  the 
sand,  with  the  view  to  catch  a  vision  of  the  true  principles  of 
the  quadrature.  He,  doubtless,  perceived  that  his  method  of 
deducing  a  progression  from  his  geometric  formula,  would  ne- 
cessarily lead  to  discrepancy  and  error,  and  that  there  existed, 
somewhere  in  his  device,  an  assumption  based  upon  false  and 
erroneous  premises.  But  we  do  not  learn  that  he  attempted  to 
degrade  the  question  of  the  quadrature,  or  to  call  those  fools  who 
might  attempt  to  improve  upon  what  he  had  done,  (as  if  wisdom 
was  to  die  with  him,)  as  has  been  the  case  with  some  others  ; 
nor  do  we  find  that  he  was  bold  enough  to  deny  the  existence  of 
what  he  did  not  comprehend,  or  to  allege,  as  a  teacher  of  science, 
that  error  is  equally  beneficial  with  truth. 

I  shall  not  pretend  or  allege,  that  the  popular  error  in  respect 
to  the  quadrature,  arises  from  any  defect  in  rectilinear  geometry, 
as  that  is  perfect  in  its  legitimate  sphere ;  nevertheless,  all  ex- 
perience has  shown  that  the  equation  of  curves,  as  well  as  of 
the  laws  of  force  and  motion  operating  in  planetary  orbits,  is 
properly  transferred  to  the  transcendentalism  of  logarithms :  Nor 
has  the  necessity  of  a  logarithmic  equation  been  overlooked  in 
attempts  to  develop  the  true  quadrature ;  and,  indeed,  the  most 
approved  method  of  extending  what  is  conceived  to  be  an  ap- 


OF    THE    CIRCLE.  31 

proximation,  is  based  upon  a  logarithmic  series.     But  I  am  con- 
strained to  allege,  that  in  lieu  of  adapting  a  logarithmic  series 
to  a  true  equation  of  the  circle,  they  have  only  adapted  it  to 
those  errors  incident  to  their  supposed  approximation,  and  which, 
of  course,  only  serves  to  confirm  the  error.    For  they  have  wholly 
disregarded  the  varying  capacities  of  polygonal  measures,  other- 
wise than  that  arising  from  the  presumption  that  the  perimetrical 
capacities  were  constantly  increasing  from  the  trigon  or  square, 
to  the  periphery  of  the  circle.     And  consequently  they  have  given 
the  periphery  of  the  circle   a  greater  capacity  than  it  actually 
possesses ;  for,  if  there  be  other  extraordinary  capacities  to  be 
consulted  or  considered,  than  those  of  the  perimeters  only,  in  a 
polygonal  progression  from  the  trigon    (or  from  the  square)   to 
the  circle,  it  must  be  manifest,  that  the  final  capacity  of  the  peri- 
meter of  the  circle,  must  depend  upon  a  proper  equation  of  the 
capacities  of  polygonal  measures,  throughout  the  whole  progres- 
sion.    If,  however,  we  attempt  an  approximation  by  means  of  a 
logarithmic  series,  it  is  certainly  of  importance  that  such  series 
should  be  adopted  as  will  lead  to  righ:  conclusions  ;  nor  should 
some  solitary  coincidence  in  the  application  of  numbers,  (sup- 
posed to  have  some  analogy,)  be  sufficient  to  establish  the  cor- 
rectness of  such  logarithmic  series ;  as  may,  perhaps,  have  been 
the  case  in  reference  to  a  series  carried  on  by  an  alternate  divis- 
ion by  the  odd  numbers,  upon  the  supposition  that  they  are 
equally  applicable  to  the  deflection   of  a  revolving  body  from  a 
tangent  to  its  orbit,  as  to  the  rectilinear  fall  of  bodies  near  the 
earth ;  and,  consequently,  that  they  had  a  like  application  to  the 
curve  of  the  circle ;  and  thus,  trusting  to  fallacious  appearances 
or  unfounded  analogies,  the  truth  has  been  permitted  to  suffer.  For, 
notwithstanding  the  law  of  the  rectilinear  fall  of  bodies  near  the 
earth   has  so  beautiful  and  simple   a  numerical  expression,  by 
which  law  the  sum  of  the  spaces  passed  over  in  as  many  equal 
moments  from  the  commencement  of  the  fall, is  the  square  of  the 
whole  number  of  moments,  (or  times  ;)  and  also,  by  which  the 
square  of  the  sum  of  all  the  moments   (or  times)  will   be  the 
sum  of  all  the  odd  numbers  in  their  order,  up  to  the  number  of 
moments,   (or  times,)  yet,  I  am  quite  apprehensive  that  those 
beautiful  coincidences  have  been  intruded  into  those  investiga- 
tions with  which  they  have  no  particular  concern,  and  have  in- 
duced conclusions  highly  detrimental  to  science.     It  is  not  my 
intention,  however,  to  reject  legitimate  evidence,  in  whatever 
form  it  may  be   offered ;  whether  it  be  direct  and  positive,  or 
only  presumptive  and  corroborative,  —  as   by  the  reductio  ad 
absoirdum ;  or  such  as  shall  arise  from  the  discrepancies  and 


32  ON    THE    QUADRATURE 

errors  of  the  popular  methods ;  and  I  shall  even  attempt  to  in- 
duce assent  to  the  truth  of  my  conclusion,  from  the  beauty  and 
harmony  of  its  analogies  and  applications,  and  its  coincidences 
with  all  the  great  laws  of  nature,  which  are  the  subjects  of  nu- 
merical or  mathematical  calculation.  For,  to  determine  the  true 
measures  of  the  circle,  and  to  know  from  indubitable  proof  that 
such  determination  is  right,  is,  perhaps,  all  that  can  rationally  be 
required ;  nor  are  we  to  be  debarred  from  any  legitimate  evi- 
dence that  may  arise  to  sustain  us  in  our  knowledge  of  the 
truth. 

My  purpose  is,  to  prove  to  the  satisfaction  of  the  world,  that 
the  circumference  of  the  circle,  whose  diameter  is  unity  or  1,  is 
3.1748020,  or  the  third  or  cube  root  of  32,  and  hence  that  the 
area,  or  one  fourth  of  the  circumference  is  .7937005,  or  the  cube 
root  of  .5,  in  lieu  of  the  popular  series,  —  .78539816339,  &c., 
&c. ;  thus  making  the  cube  or  third  power  of  the  diameter  of 
the  circle,  twice  the  third  power  of  one  fourth  of  the  circumfer- 
ence;  and,  consequently,  that  the  ratio  of  the  diameter  to  the 
circumference  is  commensurable  in  the  third  powers  and  roots 
of  definite  numbers. 

It  is  agreed  that  the  surfaces  of  solids  are  as  their  bulks : 
That  the  product  of  one  sixth  of  the  surface  of  any  regular  solid 
by  the  diameter  of  its  inscribed  sphere,  gives  the  bulk  or  con- 
tents: That  the  bulks  of  solids  increase  or  decrease  in  a  tripli- 
cate ratio,  to  that  of  any  given  linear  measure  of  the  solid  : 
That  the  surface  of  a  solid  increases  or  decreases  by  a  duplicate 
ratio  to  that  of  its  linear  measure :  And  also,  that  the  ratio  of 
the  increase  of  surface  to  that  of  bulk,  is  as  that  of  the  diame- 
ter of  the  orbits  to  the  periods  of  the  planets. 

It  has  been  well  demonstrated,  that  the  bulk  of  the  sphere  is  two 
thirds  that  of  its  circumscribing  cylinder ;  hence,  by  my  determi- 
nation, the  bulk  of  the  prime  sphere,  or  sphere  whose  diameter 
is  unity,  is  the  third  root  of  the  fraction  .148148,  ad  infini- 
tum  ;  and  the  third  power  of  its  surface  is  32.  These  pro- 
portions between  the  sphere,  cylinder  and  cube  certainly  pre- 
sent many  beautiful  coincidences  and  reciprocities,  not  only 
in  the  economy  of  numbers,  but  in  the  perfect  adaptation  and 
fitness  of  things  in  respect  to  surfaces  and  solids,  or  their  ra- 
tional admeasurement,  which  was  anciently  thought  so  desir- 
able for  the  furtherance  of  science,  as  to  be  earnestly  attended  to. 
Nevertheless,  the  beauty,  harmony  and  reciprocity  in  the  ap- 
plication, is  but  presumptive  evidence,  and  is  not  to  be  consid- 
ered sufficient,  for  satisfying. the  abstract  geometer  or  mathema- 
tician,— which  is  to  be  done.  But  as  it  is  my  intention  to  be  tried 


OF    THE    CIRCLE.  33 

by  my  peers,  I  shall  address  myself  to  them,  in  support  of  my 
cause,  in  language  as  intelligible  to  all  as  I  am  capable  of  using, 
or  as  circumstances  will  admit,  scrupulously  avoiding  those  meth- 
ods of  investigation  which,  to  a  great  portion  of  mankind,  are 
dark  and  mysterious,  and  absolutely  repulsive  and  terrific,  like 
the  methods  by  fluxions  and  the  calculus.  For  it  is  my  in- 
tention to  test  the  question  whether  the  great  mass  of  mankind 
are  capable  of  judging  for  themselves  in  respect  to  physical 
science,  or  whether  they  are  forever  to  depend  upon  the  few  pro- 
foundly learned,  for  an  implicit  faith  instead  of  actual  knowl- 
edge. And  this  being  my  purpose,  perhaps  none  will  expect 
me  to  give  heed  to  those  methods  of  investigation  which  have 
been  so  little  understood,  save  by  the  profoundly  learned  ;  and 
which,  although  carried  to  such  an  awful  height  as  to  be  beyond 
the  ken  of  mortals  in  general,  have  never,  as  yet,  perhaps,  dis- 
closed or  developed  a  single  physical  truth  worthy  of  science. 
The  method  which  I  have  adopted  for  comparing  polygonal 
quantities,  with  a  view  to  some  unity  of  purpose,  varies  some- 
what, in  many  respects,  from  the  popular  methods  ;  but  nev- 
ertheless, retains  a  sufficient  similarity  to  enable  us  to  make 
all  proper  comparisons  between  the  methods  and  results  ar- 
rived at.  And  as  it  is  my  intention  to  avail  myself,  in  behalf 
of  my  views,  of  the  discrepancies  and  errors  incident  to  the 
popular  methods  and  results  ;  and  to  show  that  they  are 
wholly  inadequate  to  equate  the  varying  capacities  of  polygonal 
measures,  it  will  be  my  endeavor,  so  far  as  I  may  refer  to  the 
popular  methods,  and  the  results  arising  therefrom,  with  a  view 
to  a  just  comparison,  faithfully  to  translate  the  same  as  nearly 
to  some  unity  of  purpose,  as  the  idiom  or  nature  of  the  case  will 
admit;  and  to  give  full  credit  to  the  authors,  in  the  use  of  the 
general  appellation  of  popular  methods  or  popular  determination. 
It  is  often  alleged  by  authors,  (and  perhaps  with  too  much 
emphasis  or  technicality,)  that  the  circle,  or  any  given  polygon, 
is  properly  the  area  or  surface;  but  that,  in  the  common  ac- 
ceptation, the  periphery,  or  perimeter  alone  is  called  the  circle, 
or  polygon ;  and  I  must  confess,  that,  for  most  purposes,  I  am 
inclined  to  the  common  acceptation ;  for  the  reason  that  it  is  the 
linear  quantities  which  must  determine  the  numeral  amount  of 
area  or  surface,  and  not  the  converse.  The  ratio  between  a  line 
which  forms  a  perfect  circle,  and  its  greatest  rectilinear  extension, 
(called  the  diameter,)  is  what  has  been  so  long  and  so  anxiously 
sought;  and  if  such  ratio  be  obtained,  we  shall  have  no  difficulty 
in  assigning  the  ratio  of  the  area  to  that  of  the  square,  whose  area 
is  called  unity ;  whether  such  area  of  the  square  be  merely  con- 
ventional, as  being  most  convenient,  as  a  standard,  or  whether 
5 


34  ON    THE    QUADRATURE 

it  is  founded  in  the  fitness  of  things ;  and  which,  from  necessity, 
has  induced  its  adoption  as  the  standard.  And  it  certainly  is 
more  proper  to  speak  of  the  varying  capacities  of  linear  meas- 
ures for  bounding  area  or  surface,  than  to  speak  of  the  capacities 
of  the  areas  or  surfaces  themselves. 

We  very  properly  speak  of  the  circle,  the  square,  the  octagon, 
&c.,  without  regard  to  their  numerical  dimensions;  and  such 
should  always  be  the  technical  consideration  of  all  regular 
polygons ;  namely,  that  a  regular  polygon,  having  any  given 
number  of  sides,  is  properly  one  and  the  same  polygon,  what- 
ever may  be  its  numerical  dimensions.  But  as  a  polygon  must 
often  be  compared  with  the  same  polygon,  possessing  different 
numerical  dimensions,  it  will  often  be  the  most  convenient 
mode  of  comparison,  to  consider  the  same  polygons,  when  ex- 
pressed by  different  numerical  dimensions,  as  being  like  poly- 
gons ;  that  is,  one  is  like  the  other.  Thus,  in  the  case  of  the  prime 
circle,  or  circle  whose  diameter  is  1,  the  same  polygon  (of  what- 
ever number  of  sides)  may  be  inscribed  within,  or  circumscribed 
about  the  circle ;  and  the  inscribed  polygon  may  be  said  to  be 
like,  or  the  same  polygon  with,  the  one  which  circumscribes 
the  circle.  And  the  importance  of  considering  the  identity  or 
sameness  of  a  regular  polygon  to  depend  upon  the  number  of 
its  sides,  and  not  at  all  upon  its  numerical  dimensions,  in 
our  investigations  of  the  quadrature,  is,  perhaps,  much  greater 
than  might  generally  be  supposed  ;  and  hence  I  shall  en- 
deavor to  enforce  the  necessity  of  such  consideration  in  those 
places  where  a  different  consideration  must  necessarily  lead  to 
discrepancy  and  error.  Thus,  calling  the  circle  whose  diame- 
ter is  unity,  the  prime  circle,  any  circumscribing  polygon  with  its 
like  inscribed  polygon,  forms  a  prime  polygon,  and  is  properly 
to  be  considered  as  one  and  the  same  polygon,  possessing  differ- 
ent numerical  dimensions  ;  the  circumscribed,  being  called  the 
major  polygon,  and  the  inscribed  being  called  the  minor  poly- 
gon. And  in  what  numerical  point  or  position  the  perimeters 
of  a  major  and  minor  polygon  shall  actually  coincide,  and  there- 
by form  the  periphery  of  the  prime  circle,  is  the  question  to  be 
determined.  That  such  periphery  exists,  is  a  self-evident  pro- 
position ;  nor  is  it  in  any  way  material,  whether  we  conceive  it 
to  be  composed  of  the  perfect  coincidence  of  a  major  and  minor 
perimeter,  or  whether  we  conceive  it  to  lie  between  the  utmost 
major  and  minor  perimeters ;  it  is  sufficient  for  us  that  it  is  but 
a  boundary,  the  thickness  of  which  we  can  no  more  comprehend 
than  that  of  the  infinite  divisibility  of  matter.  Nevertheless,  as 
we  have  all  along  conceived  the  major  and  minor  perimeters  to 
be  progressing  towards  the  periphery  of  the  circle,  perhaps  our 


OF    THE    CIRCLE.  35 

utmost  conception  must  be,  that  the  periphery  of  the  prime  circle 
lies  between  the  perimeters  of  the  utmost  major  and  minor  poly- 
gons. And  hence  the  utmost  of  our  researches  in  respect  to  the 
quadrature,  can  extend  no  further  than  that  of  finding  the  utmost 
means,  (either  equal  or  proportional,)  between  two  approxima- 
ting quantities  or  numeral  positions.  This,  however,  is  quite 
sufficient ;  for  geometric  points  and  lines,  having  no  thickness, 
the  utmost  mean  between  two  approximating  points  or  lines  is 
but  the  actual  coincidence  of  those  points  or  lines ;  and  all  must 
necessarily  lie  in,  and  be  expressed  by  the  same  root  or  power. 
It  may,  nevertheless,  in  the  course  of  the  work,  often  be  more  in 
accordance  with  our  habits  of  thinking,  to  conceive  certain  coin- 
cidences to  operate  as  a  merger  or  extinguishment  of  a  part  of 
the  coinciding  quantities,  as  in  the  case  of  diametric  quantities, 
for  instance ;  for  whilst  each  polygon  of  sides  is  conceived  to 
have  two  diametric  quantities,  (namely,  the  diameters  of  its  in- 
scribed and  circumscribed  circles,)  the  circle  is  generally  con- 
ceived to  have  but  one  diametric  quantity. 

Nevertheless,  it  may  often  be  necessary,  by  way  of  compari- 
son and  explanation,  to  consider  the  circle  in  the  nature  of  a 
polygon  of  sides,  and  possessing  like  quantities  as  that  of  an 
inscribed  and  circumscribed  diameter ;  and  in  case  the  prime 
circle  is  to  be  considered  as  the  utmost  polygon,  partaking,  as 
well  of  the  properties  of  the  minor  as  of  the  major  polygons,  it 
may  also,  for  the  purposes  of  comparison,  be  conceived  to  have 
a  major  and  minor  perimeter,  and  a  major  and  minor  area ;  in 
which  case,  the  major  area  will  be  numerically  equal  to  one 
fourth  of  the  major  perimeter,  and  one  fourth  of  the  minor  perim- 
eter will  be  conceived  to  be  equal  to  a  minor  area  of  twice  the 
number  of  sides ;  for  such  is  numerically  the  utmost  of  our  con- 
ception in  respect  to  a  minor  perimeter  and  its  accompanying 
area.  And  hence,  so  long  as  we  retain  the  conception  of  a 
minor  perimeter,  so  long  we  retain  the  conception  of  a  minor 
area  of  twice  the  sides. 

Now  to  obviate  the  difficulty  in  obtaining  a  satisfactory  result 
to  the  mind,  arising  from  such  inevitable  conceptions,  as  well  as 
to  determine  the  numerical  points  or  position  in  which  the  major 
and  minor  perimeter  of  a  prime  polygon  actually  coincide  and 
form  the  periphery  of  the  prime  circle,  it  will  be  necessary,  lest 
we  mistake  the  ratio  of  progression  of  the  minor  perimeter  to 
that  of  the  major,  that  we  so  prepare  our  work  as  to  enable  us 
measurably  to  comprehend  the  end  from  the  beginning,  by  the 
adoption  of  such  a  course  as  will  furnish  a  train  of  such  absolute 
and  conclusive  coincidences  of  the  ratios  of  progression  of 
polygonal  quantities,  as  inevitably  direct  to,  and  conclusively 


36  ON    THE    QUADRATURE 

determine  the  numerical  point  or  position  in  which  the  major 
and  minor  perimeters  of  the  utmost  prime  polygon  must  actually 
coincide.  And  as  the  numeral  value  of  the  minor  perimeter 
of  any  prime  polygon,  is,  to  the  major  perimeter,  as  the  diameter 
of  the  inscribed  circle  of  a  like  polygon  is  to  the  diameter  of  the 
circumscribed  circle  of  such  polygon,  I  have  sought  to  obtain 
such  trains  of  absolute  coincidences,  (and  consequently  the  coin- 
cidence of  the  major  and  minor  perimeter  of  the  utmost  prime 
polygon,)  by  a  comparison  of  the  class  of  prime  polygons  with 
that  class  of  polygons  whose  respective  areas  are  0.5,  or  half  of 
unity,  which  class  of  polygons,  in  contradistinction  from  the 
prime  polygons,  I  have  thought  proper  to  denominate  primal  poly- 
gons; and  the  four  denotations  by  means  of  letters  of  the  alpha- 
bet, by  which  such  comparison  is  mostly  made,  I  shall  call  prime 
denotations ;  and  inasmuch  as  a  full  comparison  must  necessarily 
extend  to  polygons  of  different  numerical  dimensions,  from  those 
of  the  prime  and  primal,  other  denotations  may  be  appended, 
denominated  auxiliary  denotations ;  all  of  which  will  be  suffi- 
ciently explained  for  use,  in  the  appropriate  place. 

Perhaps  it  will  not  be  inappropriate  here,  to  give  some  intro- 
ductory hints  at  what  will  appear  far  more  conspicuous  in  the 
work,  namely :  the  admirable  adaptation  of  the  prime  and  pri- 
mal polygons  to  an  investigation  of  the  quadrature ;  not  only 
from  the  great  number  of  actual  coincidences  so  readily  obtain- 
ed in  the  ratios  of  progression  of  polygonal  quantities,  which  so 
satisfactorily  dictate  the  numeral  position  of  the  final  coinciden- 
ces sought ;  but  from  the  manner  in  which  those  progressive 
ratios  unfold  to  us  the  varied  and  revolving  capacities  of  poly- 
gonal measures,  while  performing  the  quadrant  of  an  orbit,  (or, 
if  we  please,  the  two  octants  of  the  quadrant,)  thereby  constantly 
suggesting  both  the  necessity  and  the  manner  of  equating  those 
revolving  capacities. 

Thus  the  diameter  of  the  prime  circle  is  the  inscribed  diame- 
ter of  all  major  polygons,  and  the  circumscribed  diameter  of  all 
minor  polygons ;  making  unity  an  important  point  in  this  par- 
ticular, but  mainly  from  the  fact  that  the  prime  square  possesses, 
in  and  of  itself,  important  coincidences  situated  in  unity,  thereby 
making  the  square  the  proper  polygonal  standard.  Thus  the 
diameter  of  the  inscribed  circle  of  the  square,  and  one  fourth  of 
the  perimeter  of  the  same  square,  are  always  numerally  equal ; 
and  when  situated  in  unity,  each  is  necessarily  equal  to  the  area; 
each  of  which  is  then  numerically  equal  to  the  diameter  of  the 
circumscribing  circle  of  the  primal  square,  or  square  whose  area 
is  half  of  unity,  which  circumstance  becomes  important  in  our 
investigations ;  while  one  fourth  of  the  perimeter  of  the  minor 


OF    THE    CIRCLE.  37 

square  is  equal  to  the  diameter  of  the  inscribed  circle  of  the  pri- 
mal square,  and  is  also  equal  to  the  area  of  the  minor  octagon  ; 
— thus  making  the  square  (namely,  the  polygon  denoting  unity 
amongst  the  polygons)  emphatically  the  standard  from  which 
ratio  and  proportion  are  properly  conceived  to  emanate,  and  with 
which  they  must  of  necessity  be  compared.  But  in  all  prime  po- 
lygons below  the  square,  (namely,  of  more  sides  than  the  square,) 
which  we  are  able  specifically  to  investigate,  one  fourth  of  the  pe- 
rimeter is  less  than  the  diameter  of  the  circumscribing  circle  of  a 
like  primal  polygon  ;  and  by  the  popular  methods  this  variation 
is  increased  in  the  whole  distance  from  the  square  to  the  circle  ; 
thereby  making  the  capacity  of  the  perimeter  of  the  circle  too 
great,  and  that  of  the  diameter  too  small. 

Now  this  variance  between  one  fourth  of  the  perimeter  of  a 
major  polygon  and  the  diameter  of  the  circumscribing  circle  of 
a  like  primal  polygon,  which  is  proportionally  the  same  between 
one  fourth  of  the  perimeter  of  a  minor  polygon  and  the  diameter 
of  the  inscribed  circle  of  a  like  primal  polygon,  (which  variance 
is  nothing  in  the  square  and  circle,  and  is  at  its  maximum  in  the 
octagon,)  is  always  proportionally  equal  to  twice  its  equation ; 
and  to  denote  such  equation,  a  sign  or  symbol  will  be  adopted 
and  explained.  Again,  a  comparison  of  the  prime  and  primal 
polygons  is  well  calculated  to  disclose  certain  facts  and  phenom- 
ena in  polygonal  economy,  which  the  popular  methods  seem  not 
to  have  heeded  or  to  have  made  any  account  of, — and  among 
others,  the  transposition  of  polygonal  quantities  in  unity,  or  in 
the  prime  square,  while  passing  from  the  trigon  to  polygons  sit- 
uated below  the  square, — which  phenomenon  certainly  deserves 
consideration,  and  will  receive  it,  by  special  reference,  in  due  time 
and  in  the  proper  place. 

And  it  is  possible  that  a  notice  and  consideration  of  this 
transposition  of  polygonal  quantities  in  unity,  or  the  prime 
squ archly"  have  suggested  some  defect  in  the  popular  mode 
of  approximating  by  means  of  a  logarithmic  series;  that  is,  the 
method  of  obtaining  constant  dividends  and  divisors,  and  a 
disregard  of  such  transposition,  may  have  afforded  some  of  those 
geometrical  reasons  which  are  said  to  exist,  why  "  no  ra- 
tional proportion  of  the  diameter  to  the  circumference  of  the 
circle  is  possible."  This  transposition  of  polygonal  quantities 
in  the  major  square,  is  necessarily  such,  that,  for  the  obtaining 
of  one  fourth  of  the  major  octagon,  the  dividend  and  divisor  may 
always  be  obtained  from  the  same  nominal  terms  of  the  square, 
as  from  the  trigon,  in  deducing  one  fourth  of  the  perimeter  of  the 
major  hexagon  ;  out  as  in  any  polygon  below  the  square,  the 
relative  situations  of  the  terms  are  transposed  from  what  they  are 


38  4  ON    THE    QUADRATURE 

in  the  trigon,  it  may,  at  least,  be  questioned,  whether  there 
should  not  be  some  transposition  of  terms  in  polygons  be- 
low the  square,  from  which  the  requisite  dividends  and  divi- 
sors should  be  obtained.  Nevertheless,  such  transposition  has 
not  been  heeded  in  the  popular  methods.  It  will,  however,  be 
quite  manifest,  that  dividends  and  divisors  obtained  from  any 
given  polygon  below  the  square,  for  deducing  one  fourth  of  the 
circumference  of  a  major  polygon  of  twice  the  sides,  may  be 
obtained  of  like  nominal  quantities  of  the  square  for  deducing 
one  fourth  of  the  perimeter  of  the  major  octagon  ;  but  not  from 
the  trigon,  in  deducing  one  fourth  of  the  perimeter  of  the  major 
hexagon.  The  consequence  then  becomes  inevitable,  that  if  we 
proceed  to  obtain  our  constant  dividends  and  divisors  from  no- 
minal terms,  the  same  as  or  like  those  we  use  in  obtaining  one 
fourth  of  the  perimeter  of  the  major  hexagon  and  octagon,  the 
divisor  becomes  too  large  for  the  dividend,  and  consequently 
the  quotient  will  be  too  small ;  and  the  error  will  continue  to 
increase,  by  infinitesimal  quantities,  to  the  final  determination, 
producing  some  strange  discrepancies  near  the  final  determina- 
tion ;  —  as  that  of  forming  premature  coincidences  and  -again 
breaking  up  and  dispersing  them  ;  while  other  quantities,  equally 
entitled  to  coincidence,  are  not  permitted  at  all  to  coincide ; 
and  in  the  final  result,  those  quantities  which,  from  their  ratios 
of  progression,  seemed  all  along  destined  to  form  one  general 
coincidence  in  the  circle,  are  scattered  into  various  and  diverse 
coincidences ;  and  not  a  single  term  or  polygonal  quantity  is 
permitted  to  remain  in  that  numeral  point  or  position  which 
seemed  destined,  from  every  law  of  polygonal  progression,  to  be 
the  point  of  coincidence  for  all  the  progressive  quantities. 
Thus  one  fourth  of  the  periphery  of  the  prime  circle  will  be 
.7853981,  &c.,  and  the  diameter  of  the  circle  whose  area  is  0.5, 
will  necessarily  be  expressed  by  .797892,  &c.,  and  the  three 
points  of  equation  which  should  have  directed  to  the  true  place, 
are  caused  to  coincide  in  .791620  ;  namely  :  .002080  below  the 
true  place. 

Because  the  product  of  one  fourth  of  the  perimeter  of  any 
given  polygon  X  the  diameter  of  the  inscribed  circle  of  such 
polygon  gives  the  area  of  such  polygon,  consequently,  for  the 
same  cause,  the  area  and  one  fourth  of  the  perimeter  of  any 
given  polygon,  the  diameter  of  whose  inscribed  circle  is  unity, 
are  numerically  equal ;  and  hence  it  is  equally  well,  and  vastly 
more  convenient  in  our  investigations,  to  adopt  one  fourth  of  the 
perimeter  as  the  standard  circumferential  measure,  in  lieu  of  the 
entire  perimeter. 

I  shall  plead  no  excuse  for  the  desultory  manner  in  which  I 


OF    THE    CIRCLE.  39 

shall  treat  the  subject  of  the  quadrature ;  it  being  my  design 
solely  to  furnish  the  evidence,  both  in  quantity  and  quality,  to 
satisfy  the  candid  inquirer  of  the  correctness  of  my  solution  of 
this  important  problem.  Hence,  after  setting  forth  an  epitome 
of  the  prime  table,  as  a  table  of  reference,  and  giving  some  ex- 
planation of  the  few  signs,  characters,  and  denotations  used  in 
the  work,  I  shall  proceed  to  present  my  evidence  in  such  a  way 
and  manner  as  I  may  conceive  will  elicit  the  most  thorough  and 
thoughtful  investigation  from  the  inquirer.  And  conceiving  the 
whole  subject  to  be  wrapt  up  in  consequents  or  corollaries,  flow- 
ing directly  from  well  demonstrated  principles  which  need  not  to 
be  repeated,  the  bare  suggestion  of  which,  by  way  of  allegation, 
being,  in  most  cases,  a  sufficient  reference  to  the  demonstrated 
principles  from  which  they  emanate  or  flow,  at  least  by  the  aid  of 
a  little  reflection,  and  that  correct  reasoning  of  which  Legendre 
speaks, —  I  shall  not,  in  general,  do  more  than  to  allege  such 
demonstrated  truths,  and  such  consequences  as  they  inevitably 
dictate.  And  in  this  respect,  I  may  sometimes  become  quite 
prolific  in  trains  of  such  correlative  matter  as  I  may  conceive 
will  throw  light  upon  the  subject ;  for  it  is  my  intention  to  fur- 
nish indubitable  proof  of  the  correctness  of  my  determination, 
by  showing  "  that  however  we  combine  our  evidence,  provided 
only  our  reasoning  be  correct,  the  results  we  obtain  will  always 
be  perfectly  accurate."  The  limits,  however,  assigned  to  the 
work,  will  admit  but  a  very  small  portion  of  the  evidence 
which  I  have  examined  in  favor  of  my  determination  ;  nor  am 
I  confident  that  the  evidence  which  I  shall  offer,  will  be  a  judi- 
cious selection  from  that  which  I  have  examined. 

I  am  aware  of  the  ordeal  which  the  popular  determination 
of  the  quadrature  of  the  circle  has  passed ;  and  of  the  fortified 
position  in  which  it  now  rests,  as  in  the  citadel  of  the  world's 
last  hope ;  and  that,  were  it  not  for  the  eternity  and  immutability 
of  the  principles  of  ratio  and  proportion,  the  popular  error 
must,  long  ere  this,  have  been  converted  into  a  mathematical 
truth.  And  it  having  been  thoroughly  tested  by  fluxions  and  the 
calculus,  (those  methods  which  appear  to  me  far  better  calculat- 
ed for  the  equation  of  human  errors,  than  for  disclosing  mathe- 
matical truths,)  the  learned  have  in  general,  become  satisfied 
that  the  force  of  nature  can  no  further  go  ;  and  hence,  the  ques- 
tion is  sure  to  be  put  to  the  modest  inquirer,  in  respect  to  a  true 
solution  of  the  quadrature,  "  whether  he  has  examined  the  sub- 
ject by  means  of  the  fluxions  and  the  calculus;"  and  still  we 
find  it  alleged  in  Rees'  Cyclopaedia  that  the  hopes  of  mathema- 
ticians, which  were  revived  in  respect  to  a  true  solution  of  the 
quadrature,  upon  the  invention  of  fluxions  and  the  calculus, 


40  ON    THE    QUADRATURE 

were,  at  length,  abandoned  in  despair,  after  exhausting  every 
possible  device  by  means  of  those  methods,  but  without  success; 
and  hence  a  true  solution  is  now  declared  to  be  impracticable  ; 
and  geometry  is  charged  with  having  furnished  reasons  which 
justify  such  declarations;  but  I  have  certainly  understood  legiti- 
mate geometry  quite  differently.  But  the  problem  of  the  quadra- 
ture ought  not  to  be  settled  by  convention,  although  I  am  aware 
that  important  questions  in  respect  to  science  have  too  often 
been  settled  in  that  way ;  as  in  case  of  the  theory  of  universal 
gravity ;  and  in  case  of  the  laws  by  which  gravity  operates  in 
respect  to  distance. 

If,  however,  the  popular  determination  of  the  quadrature,  and 
also  the  Newtonian  theory  and  law  of  gravity,  are  forever  to 
remain  as  the  world's  best  hope  in  respect  to  a  sure  foundation 
upon  which  to  base  their  scientific  researches,  then,  to  be  sure, 
astronomers  must  have  an  onerous  task  to  perform,  in  which 
one  equation  only  serves  to  beget  other  supposed  variances  and 
aberrations,  and  so  on,  ad  infinitum ;  in  which  case,  the  toils 
and  labors  of  astronomers  will  become  perplexing,  and,  like  the 
labors  of  Sysyphus,  never-ending. 

As  a  matter  of  reference,  I  have  thought  proper  to  close  this 
section  with  an  epitome  of  two  tables ;  the  first,  or  prime  table, 
being  formed  by  constantly  multiplying  unity,  and  each  suc- 
cessive product  obtained,  by  the  sixth  root  of  .2  in  ascending,  or 
by  the  sixth  root  of  .5  in  descending.  Hence,  if  the  natural 
numbers  are  made  exponents  to  the  terms,  both  in  ascending 
and  in  descending,  any  two  terms  denoted  by  the  same  number, 
will  be  reciprocals  of  each  other ;  the  term  denoted  by  2.,  in  the 
descending  column,  would  be  the  same  which  I  have  so  con- 
stantly denoted  by  £,  in  the  course  of  the  work,  and  is,  accord- 
ing to  my  idea,  the  area  of  the  circle  whose  diameter  is  unity. 
And  in  accordance  with  such  conclusion,  if  the  diameter  of  the 
sphere  be  taken  in  any  term  of  the  table,  or  if  the  diameter  of 
the  circle  be  taken  in  any  term  of  the  table,  the  third  power  or 
cube  of  the  surface  of  such  sphere,  or  the  third  power  or  cube  of 
the  circumference  of  such  sphere,  or  of  such  circle,  will  be  ex- 
pressed in  terms  of  the  table,  if  the  terms  are  sufficiently  ex- 
tended, as  they  may  be  at  pleasure.  And  the  third  power  or 
cube  of  the  surface  of  the  circumscribing  cylinder  of  such 
sphere,  may  be  found  expressed  in  terms  of  the  second  table,  in 
case  the  terms  be  sufficiently  extended. 


OF    THE    CIRCLE. 


41 


FKlMh 
1. 

TAUt-JK. 
1. 

»J1,UU1N. 

.16666 

U  lAJDJUJBj. 

6. 

.890898 

1.122462 

.209986 

4.762203 

.793700 

1.259921 

.264566 

3.779763 

.707106 

1.414213 

.3333 

3. 

.629960 

1.587401 

.419973 

2.381101 

.561231 

1.781786 

.529133 

1.889881 

.5 

2. 

.666 

1.5 

.445449 

2.244924 

.839946 

1.190550 

.396850 

2.519840 

1.058265 

.944940 

.353553 

2.828426 

1.333 

.75 

.314980 

3.174802 

1.679888 

.595274 

.280615 

3.563572 

2.116531 

.472467 

.25 

4. 

2.666 

.375 

.222724 

4.489848 

3.359777 

.297637 

.198425 

5.039684 

4.233062 

.236233 

.176776 

5.656852 

5.333 

.1875 

.157490 

6.349602 

.140307 

7.127144 

.125 

8. 

.111361 

8.979696 

.099212 

10.079360 

.088388 

11.313704 

.078764 

12.699204 

.070153 

14.254288 

.0625 

16. 

.055680 

17.959392 

.049606 

20.158720 

.044194 

22.627408 

.039373. 

25.398416 

.035076 

28.508576 

.03125 

32. 

It  will  be  seen  that  in  the  second  table,  the  increase  and  de- 
crease from  6,  (or  surface  of  the  prime  cube,)  and  its  reciprocal, 
is  by  a  duplicate  ratio  to  that  in  the  prime  table.  If  the  diame- 
ter and  surface  of  a  sphere  be  found  in  the  prime  table,  its  bulk 
will  be  found  in  the  second  table,  in  case  the  tables  be  ex- 
tended so  as  to  give  the  required  terms. 

So  if  the  diameter  and  bulk  of  a  cylinder  be  contained  in  the 
prime  table,  its  surface  will  be  contained  in  a  term  of  the  second 
table,  &c. 

The  above  tables  are  only  intended  as  examples,  as  they  may 
be  extended  at  pleasure  ;  similar  tables  may  also  be  formed, 
applicable  to  the  measures  of  the  circle. 
6 


42  ON    THE    QUADRATURE 


SECTION    SECOND. 


Eternal  wisdom  has  so  contrived  and  adapted  the  economy 
of  numbers  to  the  human  capacity  and  the  fitness  of  things,  in  all 
those  physical  operations  which  are  subject  to  ratio  and  propor- 
tion, or  to  a  numerical  value,  that  it  would  seem  wholly  unne- 
cessary for  us  to  be  groping  in  darkness  in  respect  to  those 
things,  if  we  but  take  heed  to  that  unity  of  purpose  in  which  they 
appear  to  have  been  contrived. 

But  as  my  purpose  now  is  to  treat  of  polygonal  economy,  as 
applicable  to  the  economy  of  numbers,  I  will  at  once  suggest 
a  fitness  of  things  which  has  placed  all  proper  investigation, 
in  respect  to  polygonal  progression  and  the  quadrature  of  the 
circle,  within  very  narrow  limits,  and  which  is,  therefore,  not 
incomprehensible  to  any  inquirer. 

It  is  well  agreed  that  the  square,  the  diameter  of  whose  in- 
scribed circle  is  assumed  at  unity,  properly  denotes  unity, 
amongst  the  regular  polygons,  (and  it  is  only  of  regular  poly- 
gons which  I  treat,)  and  that  the  right  angle  is  properly  termed 
the  angle  of  unity  amongst  angles  ;  and  as  all  ratio  and  propor- 
tion emanates  from,  and  has  a  direct  reference  to  or  connection 
with  unity,  hence  the  square,  which  denotes  unity  in  so  many 
respects,  —  namely,  the  diameter  of  whose  inscribed  circle  is  1., 
whose  area  is  1.,  and  whose  side  is  equal  to  the  diameter  of  its 
inscribed  circle,  and  in  which  the  diameter  of  its  inscribed  circle 
is  equal  to  the  diameter  of  the  circumscribed  circle  of  the  square, 
whose  area  is  .5,  or  half  of  unity,  —  is  admirably  calculated  for 
a  standard,  by  which  to  compare  the  quantities  of  tne  inscribed 
and  circumscribed  polygons  of  the  circle,  whose  diameter  is  as- 
sumed at  unity. 

Hence,  in  my  direct  investigation  of  the  quadrature,  by  the 
aid  of  symbols  or  denotations,  I  shall  confine  the  subject  to  two 
classes  of  polygons,  called  prime  and  primal ;  the  prime  class 
consisting  of  the  inscribed  and  circumscribed  polygons  of  the 
circle,  whose  diameter  is  1.  —  the  inscribed  being  called  minor 
polygons,  and  the  circumscribed  being  called  major  polygons ; 
—  the  other  class  being  that  whose  respective  areas  are  .5,  or 
half  of  unity,  which  polygons  of  the  last  class  I  shall  call  primal 
polygons. 

In  the  comparison  of  the  proper  quantities  of  these-  two 
classes,  the  proper  quantities  of  a  primal  polygon  is  its  inscribed 


OF    THE    CIRCLE.  43 

and  circumscribed  diameters ;  namely,  the  diameter  of  its  cir- 
cumscribed circle,  which  will  be  denoted  by  B ;  and  the  diame- 
ter of  its  inscribed  circle,  which  will  be  denoted  by  D. 

The  proper  quantity  of  a  minor  polygon  is  one  fourth  of  its 
perimeter,  which  will  be  denoted  by  C  ;  and  the  proper  quantity 
of  a  major  polygon  is  also  one  fourth  of  its  perimeter,  which 
will  be  denoted  by  M. 

These  four  letters  are  the  prime  denotations  for  investigating 
the  quadrature ;  and  should  other  letters  or  denotations  be  add- 
ed, for  the  purpose  of  exemplification,  they  are  to  be  esteemed 
as  only  secondary  or  auxiliary. 

It  will  be  perceived,  that  in  the  prime  denotations  for  investi- 
gating the  quadrature,  I  have  scrupulously  avoided  any  denota- 
tion for  area  ;  and  this  I  have  done  in  accordance  with  what 
I  have  before  stated,  namely,  that  linear  quantities  are  the 
only  quantities  with  which  we  have  anything  to  do,  in  the  inves- 
tigation of  the  quadrature.  The  only  question  to  be  solved,  — 
the  only  mystery  in  the  case,  —  is  that  of  the  ratio  between  di- 
ameter and  circumference,  wholly  regardless  of  area,  otherwise 
than  as  a  consequent,  depending  on  the  ratio  between  the  lines 
which  measure  or  bound  area.  Hence,  if  we  obtain  the  true  ratio 
between  the  diameter  and  circumference,  there  will  be  no  question 
in  respect  to  the  true  assignment  of  area. 

The  circle,  then,  or  any  regular  polygon,  is  properly  its  diam- 
etric and  circumferential  quantities,  and  not  its  area. 

Again :  that  lines  are  the  proper  measures  of  surfaces  or  areas, 
and  not  areas  of  lines,  is  too  manifest  to  require  comment,  not- 
withstanding the  popular  attempts  to  obtain  the  quadrature  seem 
rather  adverse  to  my  notions ;  nevertheless,  the  ill  success  at- 
tending that  method,  has  not  proved  the  legitimacy  of  the 
method.  Legitimate  geometry  does  not  geometrize  upon  areas, 
but  upon  lines. 

Geometry,  however,  seeks  for  three  or  more  lines,  from  which 
to  deduce  ratio  and  proportion ;  but  it  never  seeks  to  proportion 
lines  by  means  of  areas,  by  a  disregard  of  those  lines. 

Thus  the  business  of  legitimate  geometry  is,  to  proportion 
lines  to  each  other ;  and  in  this  it  does  not  at  all  regard  the 
varied  and  varying  capacities  of  polygonal  perimeters  for  cir- 
cumscribing area.  Nevertheless,  whatever  may  be  the  varied 
capacities  of  polygonal  perimeters  for  circumscribing  area,  in 
proportion  to  their  respective  diametric  quantities,  or  however 
different  that  of  the  octagon,  or  of  the  polygon  of  sixteen  sides, 
may  be  from  that  of  the  square,  we  shall  find  that  what  may 
properly  be  termed  the  capacity  of  the  perimeter,  in  proportion 


44  ON    THE    QUADRATURE 

to  the  diametric  quantity,  is  the  same  in  the  circle  as  in  the 
square. 

Now  it  has  not  wholly  escaped  the  ken  of  mathematicians 
and  geometers,  that  the  capacity  of  the  perimeter  of  the  octagon, 
for  circumscribing  area,  is  greater,  in  proportion  to  its  two  diam- 
etric quantities,  than  that  of  any  other  polygon ;  nevertheless, 
no  advantage  or  benefit  seems  to  have  been  taken  of  the  fact, 
although  it  is  of  such  vast  importance  in  a  consideration  of  the 
quadrature ;  for,  while  attempting  a  geometrical  induction  into  a 
true  polygonal  progression,  the  method  has  been  to  geometrize 
upon  areas  in  lieu  of  lines,  thus  losing  all  benefits  to  be  derived 
from  the  varying  capacities  of  polygonal  measures. 

But  areas  need  not  be  even  named  or  known,  in  a  polygonal 
progression,  except  in  the  primal  class,  where  they  silently  keep 
station  in  .5. 

Such,  then,  being  the  case  in  respect  to  the  capacities  of  poly- 
gonal measures  for  bounding  area,  geometry  will  readily  dictate 
a  polygonal  progression  from  the  major  octagon  to  the  prime 
circle,  on  true  principles  of  ratio  and  proportion,  and  in  a  way 
and  manner  that  cannot  well  be  misunderstood ;  nor  need  we 
resort  to  other  diagrams  than  such  as  are  to  be  found  in  Legen- 
dre,  and  other  works  on  geometry.  It  may  be  well  here,  how- 
ever, to  remark,  that  the  areas  of  polygons  are,  properly  speak- 
ing, proportioned  to  the  diameters  of  their  circumscribed  cir- 
cles. 

I  will  now  proceed  to  set  forth  the  signs,  characters  and  de- 
notations to  be  used  or  employed,  as  a  kind  of  analytical  machin- 
ery, in  deducing  the  quadrature,  with  the  requisite  explanations 
for  reading  the  same,  —  those  denoting  multiplication,  division, 
equality  and  proportion  being  used  in  the  common,  or  popular 
manner. 

Numerical,  or  numeral  quantity,  may  properly  be  conceived 
to  be  that  which  is  contained  between  0  and  some  given  nume- 
ral point)  (or  position,)  as  that  of  .5,  or  the  square  of  .5,  for  ex- 
ample ;  or,  numeral  quantity  may  be  contained  between  two 
points,  (or  positions,)  expressed  or  understood  ;  as  that  con- 
tained between  .5  and  1.  for  example  ;  and  such  will  be  the 
technical  meaning  given  to  the  word  point,  or  points,  when  re- 
ferring to  boundaries  of  numerical  quantities.  Thus,  we  may 
speak  of  a  root  or  power  of  a  given  point,  or  that  a  given  point 
is  a  mean  between  two  other  given  points,  &c. 

The  word  mean,  when  used  to  denote  a  point  between  two 
other  points,  when  not  represented  to  be  an  equal  mean,  is  always 
to  be  understood  as  denoting  proportional  quantity  ;  thus  2.  is  a 


OF    THE    CIRCLE.  45 

mean  between  4.  and  1. ;  so  2.  is  the  farther  of  two  means  from 
8.  to  1.,  or  the  farther  of  three  means  from  16.  to  1.,  &c. 

^  denotes  the  first  term  of  the  prime  table  below  unity ;  %,  the 
second  term,  and  so  on. 

^  This  sign,  with  a  figure  under  it,  is  used  to  denote  any 
given  number  of  terms  of  the  prime  table,  at  which  one  given 
point  may  be  situated  either  above  or  below  another  given  point ; 
thus,  for  example,  2.  is  ^  above  1.,  &c. 

p  denotes,  generally,  a  polygon,  and  may  be  read  polygon. 

b  denotes,  generally,  the  diameter  of  the  circumscribed  circle 
of  any  given  polygon,  except  when  the  area  is  .5  ;  in  which  case 
it  will  be  denoted  by  B. 

d  denotes,  generally,  the  diameter  of  the  inscribed  circle  of 
any  given  polygon,  except  when  the  area  is  .5 ;  in  which  case  it 
will  be  denoted  by  D. 

m  is  used  to  denote,  generally,  one  fourth  of  the  perimeter  of 
a  given  polygon,  except  in  case  of  a  major  polygon,  namely, 
when  the  diameter  of  the  inscribed  circle  is  1.  ;  in  which  case  it 
will  be  denoted  by  M. 

A  figure  placed  at  the  right  of  JP,  or  of  any  letter  denoting 
polygonal  quantity,  signifies  or  denotes  the  number  of  sides  of 
the  given  polygon;  thus  p  3  denotes  the  trigon,  p\  the  square, 
and  so  on.  So  b  8  or  B  8  denotes  the  octagon,  &c. 

V  is  used  to  denote  one  side  of  a  major  polygon. 

W  is  used  to  denote  one  side  of  a  minor  polygon ;  and  a 
figure  at  the  side  of  V  or  W,  denotes  the  number  of  sides  of  the 
given  polygon. 

What  I  shall  term  a  series  of  polygons,  is  one  commencing 
with  the  square  or  with  a  polygon,  the  number  of  whose  sides  is 
an  odd  number,  and  proceeding  ad  infinitum,  by  a  constant  du- 
plication of  the  number  of  sides. 

The  term  polygonal  progression,  is  used  to  denote  the  order  in 
which  the  prime  denotations  progress  through  the  major  and 
minor  polygons,  from  the  square  to  the  circle. 

A  may  be  used  to  denote  the  area  of  a  minor  polygon. 

I  will  now  proceed  to  set  forth  a  train  of  facts,  which,  in  the 
course  of  the  work,  may  become  important;  the  truth  of  them 
has  been  so  often  and  so  well  demonstrated,  and  they  are  in  and 
of  themselves  so  manifest,  —  as  may  readily  be  proved  by  any 
inquirer, —  that  it  would  be  supererogation  to  do  more  than  to 
name  or  allege  them;  and  in  doing  this  I  shall  not  attempt 
method,  but  shall  proceed  in  a  very  desultory  manner,  premis- 
ing, however,  that  the  point  denoted  by  ^,  (which  is  the  cube 
root  of  .5,)  so  constantly  occurs  as  a  mean  between  quantities 
denoted  by  two  polygonal  denotations,  which  become  proper- 


40  ON    THE    QUADRATURE 

tional  extremes  to  ^,  that  I  have  thought  proper  to  use  the  word 
adverse,  in  reference  to  such  extremes ;  as  that  they  are  adverse, 
or  one  is  the  adverse  of  the  other. 

And  first  in  the  train  of  facts,  I  will  speak  of  certain  coin- 
cidences and  reciprocities,  whereby  we  shall  find  that  we  need 
not  wander  a  great  way  from  unity  in  our  investigations  of  the 
quadrature. 

It  is  understood  that  the  area  of  any  given  major  p  is  —  M 
of  the  same  p;  also  that  C  of  any  given  minor  p  is  =  the  area 
of  a  minor  p  of  twice  the  sides. 

The  reciprocal  of  M  of  any  given  p  is  ==  d  of  a  like  p,  whose 
m  is  1. ;  and  the  reciprocal  of  C  of  any  p  is  =  b  of  a  like  p, 
whose  m  is  1. ;  and  the  reciprocal  of  the  area  of  any  minor  p  is 
=  b  of  a  jo  of  half  the  sides,  whose  m  is  1. 

When  d  of  any  p  is  the  reciprocal  of  M  of  a  like  p,  the  area 
of  such  p  is  =  d;  and  when  b  of  any  p  is  the  reciprocal  of  the 
area  of  a  like  minor  p,  the  area  of  such  p  is  =  b. 

Whenever  a  p  is  placed  in  connection  with,  or  is  compared 
with  a  p  of  half  the  sides,  it  must  always  be  understood  to 
be  a  p  of  twice  the  sides  of  another  p ;  thus  neither  the  jt?4,  nor 
any  p  of  an  odd  number  of  sides,  can  have  a  p  of  half  the 
sides.  * 

Thus  we  say  b  of  any  p  whose  area  is  =  b,  is  a  mean  between 
b  of  a  p  of  half  the  sides,  whose  area  is  =  b,  and  d  of  such  p  of 
half  the  sides,  whose  area  is  —  such  d;  hence,  such  will  con- 
tinue in  a  series,  from  the  p  8  to  the  circle. 

So  B  of  any  p  is  a  mean  between  B  and  D  of  a  p  of  half  the 
sides;  which  order  of  progression  will  also  continue  from  the  j^S 
to  the  circle. 

Because  d,  when  m  is  1.,  is  the  reciprocal  of  M,  hence,  when 
M  is  in  ^,  d  will  be  the  cube  root  of  2.,  or  ^  above  M.  So  also, 
if  C  be  in  ^,  b  will  be  the  cube  root  of  2,  or  T  above  C. 

But  ^  will  be  found  to  be  an  extremely  important  point,  in 
respect  to  polygonal  progression  by  means  of  the  prime  denota- 
tions, as  also  of  auxiliary  denotations,  however  they  be  multi- 
plied or  increased  in  numbers  ;  and  as  well  in  accordance  with 
the  popular  method  as  with  my  own,  ^  is.  me  farther  of  two 
means  from  M  to  D  ;  hence,  when  M  is  in  ^,  D  is  also.  So  ^  is 
the  farther  of  two  means  from  C,  to  B  of  twice  the  sides ;  so 
that  when  C  is  in  ^,  B  of  twice  the  sides  is  also. 

This,  then,  presents  a  consequence,  which,  perhaps,  had  better 
be  noticed  here,  namely,  if  M  and  C  coincide  in  ^,  thereby  form- 
ing M  of  the  prime  circle,  B  and  D  must  necessarily  coincide 
in  ^,  thereby  forming  D  or  B  of  the  circle ;  in  which  case,  B  of 
twice  the  sides  could  not  there  exist  in  reality.  And  it  will  be 


OF    THE    CIRCLE.  47 

fully  shown  that  if  prime  or  auxiliary  denotations,  however  nu- 
merous they  may  become,  have  their  final  place  in  £, ;  that  all 
the  denotations  (except  those  which  coincide  above  1.,)  will  also 
coincide  and  have  their  final  place  in  ^ ;  in  which  point  all  de- 
notations coinciding  below  1.  will  be  merged,  or  they  will  be  ex- 
tinguished in  the  denotations  of  the  circle. 

When  M  and  D  are  in  ^,  half  the  square  of  d,  when  m  is  1, 
is  also  in  ^.  So  when  C  is  in  ^,  half  the  square  of  &,  when  m 
is  1,  is  also  in  ^.  And  when  m  of  any  given  p  is  =  B  of  a 
like  PJ  the  area  of  such  p  is  half  the  square  of  b  of  a  like  p, 
whose  m  is  1.  So,  also,  when  m  of  any  given  p  is  =  D  of  a 
like  j9,  the  area  of  such  p  is  =  half  the  square  of  d  of  a  like  p 
whose  m  is  1. 

And  it  may  be  here  remarked,  that  the  easiest  and  simplest 
method  of  obtaining  M  8  from  terms  of  the  p  4,  would  be  the 
same  by  the  popular  method,  as  by  my  own,  —  namely,  by  di- 
viding the  product  of  M  4  by  C  4,  by  half  the  square  of  d  8 
when  m  8  is  1.  But  it  is  quite  certain,  that  such  process  will 
not  avail  any  farther  than  that  of  finding  M  8,  —  as  it  is  deter- 
mined or  shown  to  be  otherwise,  even  by  the  popular  method 
and  determination ;  for  in  the  popular  determination,  B  by  D  -f- 
half  the  square  of  6,  when  m  is  1,  gives  M,  —  that  is,  accord- 
ing to  the  final  coincidences  deduced  by  the  popular  method. 

Hence,  a  method  which  will  determine  M  8,  is  not  sure  to  de- 
termine M  of  any  p  farther  advanced  in  the  series. 

Any  given  linear  measure  of  a  given  /?,  is  always  a  mean  be- 
tween the  area  of  such  p,  (whatever  may  be  its  dimensions.)  and 
the  point  in  which  the  area  and  a  like  linear  measure  of  a  like 
py  are  = ;  thus  m  of  any  given  j»,  is  a  mean  between  the  area  of 
such  7?,  and  M  of  a  like  p,  for  the  reason  that  if  area  be  conceived 
to  be  either  increased  or  diminished,  (still  retaining  the  same 
form,)  area  will  flow  by  a  duplicate  ratio  to  that  of  any  linear 
measure  of  such  area. 

In  any  given  jo,  C  is  to  D  as  M  is  to  B,  —  hence,  C  is  to  M 
as  D  is  to  B.  But  in  the  p  3  the  relative  positions  of  C  and  D, 
and  of  M  and  B  are  different  from  what  they  are  in  any  given 
p  below  the  p  4,  or  between  the  p  4  and  circle.  Thus,  in  the  p 
3,  M  is  above  B,  and  C  is  above  D, —  but  in  any  p  between  the 
p  4  and  circle,  B  is  above  M,  and  D  above  C ;  while  in  the  p  4, 
B  and  M  are  =,  and  D  and  C  are  =. 

Hence,  if  we  take  our  dividend  and  divisor  for  obtaining  M  6, 
from  terms  of  the  p  3,  a  like  process  applied  to  the  /;  4  will  give 
M  8.  But  such  process  in  obtaining  a  dividend  and  divisor 
from  terms  of  a  major  p  of  more  sides  than  the  p  4,  will  not  be 
likely  to  give  M  of  &p  of  twice  the  sides.  Nevertheless,  any 


48  ON   THE    QUADRATURE 

process  applied  to  a  p  other  than  the  p  4,  that  will  give  M  of  a  p 
of  twice  the  sides,  will,  when  applied  to  the  p  4,  give  M  8. 

In  the  series  commencing  with  the  p  3,  M  6  is  M  of  the  first 
secondary  p  obtained  in  the  series ;  and  in  the  series  commenc- 
ing with  the  p  4,  M  8  is  M  of  the  first  secondary  p  obtained  in 
the  series ;  and  M  6  and  M  8  may  both  be  obtained  by  means 
of  the  popular  dividends  and  divisors ;  and  in  either  case,  M  of 
the  secondary  p  is  below  a  mean  between  M  and  C  of  the  p  of 
half  the  sides ;  and  the  difference  between  a  mean  between  M 
and  C  of  the  p  3  and  M  6,  is  greater  than  that  between  a  mean 
between  M  and  C  of  the  p  4  and  M  8. 

But,  by  any  system,  the  difference  between  M  and  C  of  a  jt?, 
and  M  of  a  p  of  twice  the  sides,  eventually  becomes  extinguish- 
ed ;  for  by  the  popular  method,  it  must  be  extinguished  in  the 
circle ;  and  few  things  can  be  made  more  manifest  than  that 
such  difference  is  extinguished  in  ,£,.  Neither  will  it  be  con- 
tended that  M  is  eventually  above  a  mean  between  M  and  C  of 
&p  of  half  the  sides. 

But  this  remarkable  fact  should  not  be  disregarded,  that  while 
we  find  M  in  ^  a  mean  between  M  and  C  of  a  p  of  half  the 
sides,  we  also  find,  that  if  a  supposed  progression  be  farther  con- 
tinued, (and  for  want  of  a  proper  equation  of  polygonal  capaci- 
ties, such  may  be  the  case,)  M  will  forever  after  continue  to  be 
a  mean  between  M  and  C  of  a  p  of  half  the  sides. 

Hence,  if  we  assume  a  mean  between  M  and  C  to  be  the 
mean  place  of  M  of  twice  the  sides,  and  institute  a  polygonal 
progression  upon  such  assumption,  we  shall  see  how  much  de- 
termined a  mean  between  M  and  C  is  to  coincide  with  the  other 
polygonal  quantities  in  ^. 

By  such  process  then,  the  true  and  mean  place  of  M  will  be 
found  to  coincide  in  ^,- —  and  will  remain  so  ever  after  ;  hence, 
if  they  have  not  coincided  before  arriving  at  ^,  then  after  arriv- 
ing at  ^,  the  law  of  progression  must  be  changed,  in  order  that 
the  progression  may  be  continued  farther. 

But  I  propose  to  show  that  M  becomes  a  mean  between  M 
and  C  of  a  p  of  half  the  sides,  before  we  arrive  at  ^;  and 
hence,  that  the  above  process  is  only  given  by  way  of  the  re- 
ductio  ad  absurdum. 

But  as  it  was  manifest,  that  the  difference  between  M,  and 
M  and  C  of  a  p  of  half  of  the  sides,  must  eventually  be  extin- 
guished, the  popular  methods  have  essayed  to  do  it  by  a  suc- 
cession of  dividends,  and  corresponding  divisions ;  in  which 
case,  each  successive  dividend  and  divisor  must  depend  upon 
the  correctness  or  the  incorrectness  of  the  preceding  processes. 


OF    THE    CIRCLE.  49 

Hence,  if  the  process  be  erroneous,  the  result  must  be  so  too, 
and  the  final  quantities  will  not  be  found  in  their  true  places. 

Now,  with  a  view  to  some  further  examination  of  the  popu- 
lar mode,  by  means  of  dividends  and  divisors,  I  will  simplify 
it  by  translating  it  to  a  correspondence  with  some  unity  of  pur- 
pose. 

Thus,  the  popular  method  essays  to  obtain  M  of  the  prime 
circle,  by  a  constant  division  of  the  product  of  M  by  A,  (which 
is  but  the  square  of  C,)  by  half  the  sum  of  A  and  C,  namely, 
by  an  equal  mean  between  A  and  C,  —  supposing  that  such 
process  would  serve,  from  the  p  4  to  the  circle,  to  give  M  of  a  p 
of  twice  the  sides. 

But  it  will  be  my  business  to  show  that  the  course  pursued 
conducts  truly  no  farther  than  the  rugged  octagon,  —  that  p 
whose  perimeter  has  transcendental  capacities. 

Precisely  the  same  continued  and  final  results  will  be  obtain- 
ed as  in  the  above  process,  if  we  divide  the  product  of  M  by  D, 
or  of  C  by  B,  by  half  the  sum  of  B  and  D.  Hence,  we  find 
that  B  and  D  need  not  be  excluded  from  participating  in  a 
polygonal  progression,  with  a  view  to  obtain  the  quadrature. 

So  upon  the  same  principle,  (if  correct,)  m  of  any  given  /?, 
(of  whatever  dimensions,)  -r-  by  half  the  sum  of  d  and  b  of  such 
/?,  would  give  M  of  a  p  of  twice  the  sides. 

But,  perhaps,  the  simplest  of  these  methods  is  that  in  which 
B  and  D  are  mostly  used,  —  as  by  dividing  the  product  of  M  by 
D,  —  or  C  by  B,  by  half  the  sum  of  B  and  D,  and  by  such  pro- 
cess we  certainly  obtain  M  6  and  M  8. 

Now  the  product  of  M  by  C  of  the  p  4  -f-  by  half  such  M 
and  C,  gives  M  8,  —  so  the  product  of  M  by  C  of  the  p  3  -f-  by 
half  of  such  M  and  C,  gives  M  6;  but  such  process  cannot  avail 
farther;  for  M  by  C  of  the/?  8  -H  by  half  such  M  and  C,  gives 
.783100,  —  or  even  less  than  the  popular  M  of  the  prime  circle ; 
which  intimates  very  strongly,  the  extraordinary  capacity  of  M 
8.  And  yet  nothing  is  more  manifest,  than  that  in  the  final 
result,  the  product  of  M  by  C  -^  by  half  the  sum  of  such  M  and 
C,  gives  M, —  or  M  of  twice  the  sides,  if  we  so  choose  to  con- 
sider it ;  and  if  such  progression  were  conducted  on  the  popular 
method,  the  final  M  would  be  far  below  that  of  the  popular  final 
M,  for  the  disproportion  between  the  dividend  and  divisor  would 
be  even  greater  than  by  the  popular  method.  Hence,  from  the 
p  8  to  the  circle,  there  is  required  an  equation  of  perimetric  ca- 
pacity. 

Various,  indeed,  (however  erroneous  the  process  and  results,) 
may  be  the  methods  of  conducting  a  progression  by  means  of  a 
constant  succession  of  dividends  and  divisors.     Thus,  the  pro- 
7 


50  ON    THE    QUADRATURE 

duct  of  M  by  D,  —  or  of  C  by  B  ~-  by  half  the  sum  of  C  and 
M,  will  give  the  same  continued  and  final  results,  as  if  the  pro- 
duct of  B  by  D  be  -4-  by  half  the  sum  of  such  B  and  D,  —  and 
neither  of  these  progressions,  if  thus  conducted,  will  give  the 
termini  below  £, ;  for  when  M  and  D  coincide  in  ^,  (as  they 
must  by  any  method,)  the  divisor,  namely,  half  the  sum  of  M 
and  C,  cannot  be  above  ^,  unless  we  conceive  C  to  be  above  M. 

Now  in  any  p  between  the  p  4  and  circle,  A  by  B  ~  by  half 
of  A  and  B  will  give  less,  than  when  M  by  D  is  -r-  by  half  of 
M  and  D,  —  for  the  reason  that  an  equal  mean  between  A  and 
B,  is  greater  than  an  equal  mean  between  M  and  D,  —  while  a 
mean  proportional  between  A  and  B  is  the  same  as  that  between 
M  and  D.  Nevertheless,  the  only  point  of  coincidence  of  A 
and  B,  or  of  M  and  D,  is  in  „£,,  by  any  system. 

Again,  the  product  of  C  by  D  -H  by  half  of  A  and  D,  —  is 
identical  in  result,  to  that  when  M  by  D  is  -4-  by  half  of  C 
and  B. 

Now  if  we  denote  the  difference  between  C  8  and  D  8,  —  or 
between  M  8  and  B  8,  by  J_,  —  then  the  product  of  C  8  by  D 
8  is  J_  above  the  square  of  C  8,  —  and  half  the  sum  of  A  8 
and  D  8  will  consequently  be  J  above  the  popular  divisor, 
namely,  half  of  A  8  and  C  8,  —  and  hence,  the  quotient  will  be 
J  above  the  popular  quotient,  —  that  is,  by  such  process,  M  16 
would  be  £  above  the  popular  M  16,  and  it  is  manifest  that  such 
process  would  dimmish  the  difference  between  M  and  B,  or  be- 
tween C  and  D,  while  the  popular  mode  serves  to  increase  the 
difference  between  C  and  D,  —  or  between  M  and  B,  —  or  to 
enlarge  the  quantity  denoted  by  J_,  or  cause  it  to  increase  from 
the  p  8  to  the  circle. 

Whether  we  -f-  the  product  of  C  by  D  by  half  of  A  and  D,  — 
or  the  product  of  M  by  D  by  half  of  C  and  B,  the  divisor  is 
always  to  a  mean  between  M  and  D,  as  such  mean  is  to 
the  quotient.  Hence,  where  the  divisor  becomes  the  adverse 
of  D,  the  quotient  will  be  in  ^,  —  for  a  mean  between  M  and 
D,  is  a  mean  between  the  divisor  and  quotient, —  and  ,£,  is  the 
farther  of  two  means  from  D  to  a  mean  between  M  and  D. 

If  then  a  progression  must  be  carried  on  by  means  of  con- 
stant dividends  and  divisors,  (which  is  at  least  problematical,) 
there  seems  no  good  reason  why  diametric  quantities  may  not 
share  in  the  progression,  upon  any  system  whatever.  And 
as  the  divisor  is  to  the  square  root  of  the  dividend,  as  such  root 
is  to  the  quotient,  hence,  if  M  and  D  of  any  given  p  below  the 
p  4,  being  -i-  by  half  of  such  M  and  D,  will  give  M  of  a  p  of 
twice  the  sides,  then  our  conceptions  at  a  certain  stage  of  the 
process  will  be  right ;  namely,  when  M  and  D  coincide  in  ^r 


OF    THE    CIRCLE.  51 

in  which  case  an  equal  and  a  rational  mean  between  them  would 
be  the  same  ;  and  no  further  progress  could  be  made ;  and  similar 
may  be  shown  to  be  the  results  of  other  progressions. 

Nevertheless,  all  agree  that  the  difference  between  the  di- 
visor, and  B  of  the  p  of  twice  the  sides  of  that  from  which  the 
dividend  is  obtained,  may  be  eventually  equated  to  _^_ ;  or  that 
they  may  eventually  coincide.  But  I  have  sufficiently  treated  of 
polygonal  progressions,  by  means  of  constant  or  successive  divi- 
dends and  divisors. 


SECTION    THIRD.  VJJ 

Inasmuch  as  the  point  denoted  by  £,  assumes  so  much  con- 
sequence, in  a  consideration  and  comparison  of  the  prime  and 
primal  polygons,  it  may  not  be  improper  to  note  some  few  phe- 
nomena in  respect  to  it. 

Thus,  when  m  of  any  given  p  is  made  the  adverse  of  D  of  a 
like  j9,  the  area  of  such  given  p  is  £,.  Hence,  if  m  of  a  given 
p  and  D  of  a  like  p  be  adverse,  the  product  of  one  area  by  the 
other  is  half  of  ^ ;  but  if  m  of  a  p  4  and  b  of  a  p  4  be  adverse, 
the  product  of  one  area  by  the  other  is  one  fourth  of  ^  ;  while  if 
m  of  one  circle  and  b  of  another  circle  be  adverse,  the  product  of 
one  area  by  the  other  is  half  of  £,. 

If  ^  be  the  farther  of  two  means  from  one  given  point  to 
another,  the  point  farthest  from  ^  is  half  the  square  of  the 
reciprocal  of  the  nearest  point ;  hence  M  is  always  half  the 
square  of  the  reciprocal  of  D,  A  of  B,  and  C  of  B  of  twice  the 
sides. 

The  adverse  of  M  is  always  T  below  e?,  when  m  is  1.  So  the 
adverse  of  C  is  always  "4"  below  6,  when  m  is  1.  So  also,  when 
D  of  a  p  is  the  adverse  of  M  of  a  like  JP,  the  area  of  such  p  is 
T  below  -d.  So  when  b  of  a  p  is  the  adverse  of  A  of  a  like  p, 
the  area  of  such  p  is  T  below  b. 

The  area  of  any  given  p  is  half  the  square  of  a  mean  between 
such  area  and  2. ;  hence  M  is  half  the  square  of  a  mean  between 
such  M  and  2. 

If  m  of  one  p  and  d  of  a  like  p  be  =,  the  product  of  one 
area  by  the  other  is  the  fourth  power  of  such  m  or  d. 

Twice  the  area  of  any  given  p  is  the  square  of  b  of  an  equal 
square,  with  that  of  the  given  p ;  hence,  b  of  an  equal  square, 
with  that  of  any  given  M,  is  =  m  of  a  like  p  with  that  denoted 
by  M",  whose  area  is  2. 

M  and  D  coincide  in  ^  J  hence,  when  D  is  in  ^  m  is  the 
square  of  d,  and  the  area  is  the  third  power  or  cube  of  d. 
Hence  m  of  such  p  is  the  third  root  of  the  square  of  the  area. 


52  ON    THE    QUADRATURE 

If  two  points  be  adverse,  the  reciprocal  of  either  is  '£  above 
the  other. 

The  product  of  B  by  D  of  any  p  is  half  of  b  of  a  like  p 
whose  m  is  1. 

The  square  root  of  C  is  T  above  a  mean  between  C  and 
,2, ;  so  the  square  root  of  M  is  T  above  a  mean  between  M 
and  3- 

If  b  of  one  circle  and  m  of  another,  b  of  one  p  4  and  m  of 
another,  and  b  of  one  p  3  and  m  of  another  be  =  in  one  and 
the  same  point,  the  product  of  one  area  of  the  circle  by  the 
other,  will  be  double  the  product  of  one  area  of  the  p  4  by  the 
other,  and  quadruple  the  product  of  one  area  of  the  p  3  by  the 
other. 

The  square  root  of  A  is  to  C  as  the  square  root  of  M  is 
to  M. 

Twice  the  square  of  C4is  the  equal  reciprocal  of  M4;  so 
twice  the  square  of  C  8  is  the  equal  reciprocal  of  M  8.  But  twice 
the  square  of  C  of  a  p  of  more  sides  than  the  p  8,  is  more  than 
the  equal  reciprocal  of  M. 

Hence  M  4,  plus  twice  the  square  of  C  4,  is  £  ;  and  M  8,  plus 
twice  the  square  of  C  8,  is  2.  But  in  any  p  of  more  sides  than 
the  p  8,  M  plus  twice  the  square  of  C,  is  more  than  2. 

b  of  a  major  polygon  is  the  reciprocal  of  d  of  a  like  minor 
polygon  ;  and  when  M  is  in  ^,  b  is  ^  above  B  of  a  like  j9,  and 
b  of  a  like  minor  p  is  then  ^  above  C. 

The  square  root  of  b  of  any  major/?  is  b  of  a  p  of  twice  the 
sides,  whose  area  is  =  the  area  of  the  given  prime  p. 

The  square  root  of  d  of  any  minor  p  is  d  of  a  like  p,  whose 
area  is  =  C  of  the  given  minor  p. 

D  is  the  reciprocal  of  m  of  a  like  /?,  whose  area  is  2. 

The  product  of  m  by  b  of  a  p  3,  is  ^  above  the  area ;  the  pro- 
duct of  m  by  &  of  a  jf?4,  is  T  above  the  area ;  and  the  product  of 
m  by  b  of  a  circle  is  equal  the  area. 

If  m  of  the  jf?4  be  =  M  of  any  given  p,  the  square  of  b  of 
such  p  4  is  the  reciprocal  of  half  the  square  of  b  of  the  given  p 
whose  m  is  1. 

D  of  any  given  p  is  the  reciprocal  of  b  of  a  p  4,  whose  area  is 
=  M  of  the  given  p. 

A  is  to  C  as  the  square  root  of  A  is  to  the  square  root  of  M ; 
hence,  C  is  to  M  of  twice  the  sides  as  the  square  root  of  such  C 
is  to  the  square  root  of  M  of  twice  the  sides. 

M  by  C  of  the  p  4  is  =  m  4  when  the  area  is  .5  ;  and  when  M 
and  C  of  a  p  4  are  adverse,  M  by  C  is  =  m  of  a  p  of  twice  the 
sides,  whose  area  is  .5. 


OF    THE    CIRCLE.  53 

But  M  by  C  of  any  p  between  the  p  4  and  the  p  whose  M  and 
C  are  adverse,  is  less  than  m  of  a  like  p  whose  area  is  .5. 

Half  of  the  higher  of  two  reciprocal  points  is  the  reciprocal 
of  twice  the  lower  point. 

A  mean  between  M  and  C  of  the  p  4  is  half  of  a  mean  be- 
tween b  and  d  of  the  p  4,  whose  area  is  2.,  and  the  same  with  the  p 
whose  M  and  C  are  adverse,  and  each  of  them  is  half  of  b  of  a 
p  of  twice  the  sides,  whose  area  is  2 ;  hence,  ^  above  ^  is  a 
mean  between  b  and  d  of  a  like  p  of  that  whose  M  and  C  are 
adverse,  whose  area  is  2.  Consequently,  ^  above  ^  is  b  of  a 
like  p  to  that  whose  M  is  in  ^,  whose  area  is  2. ;  and,  conse- 
quently, if  M  of  a  jo  be  below  ^,  a  mean  between  M  and  C 
will  be  less  than  half  of  a  mean  between  b  and  d  of  a  like  j9, 
whose  area  is  2. 

If  the  area  of  any  given  p  is  the  third  power,  or  cube,  of  any 
given  point,  such  area  is  a  mean  between  d  of  such  p  and  m 
of  a  like  jo,  whose  area  is  =  such  point. 

So  the  third  power  of  D  of  any  given  p  is  to  .5,  as  such  D  is 
to  M  of  a  like  p. 

So  the  third  power  of  B  of  any  given  p  is  to  .5,  as  such  B  is 
to  A  of  a  like  p. 

Hence,  the  third  power  of  d  of  a  p,  of  whatever  dimensions, 
is  to  the  area  of  such  p  as  such  d  is  to  M  of  a  like  p. 

So  the  third  power  of  b  of  a  p,  of  whatever  dimensions,  is  to 
the  area  of  such  p  as  such  b  is  to  A  of  a  like  p. 

So  also,  the  third  power  of  m  of  any  given  p  is  to  the  area  of 
such  p  as  such  m  is  to  the  reciprocal  of  M  of  a  like  p. 

The  product  of  the  area  of  any  given  p,  of  whatever  dimen- 
sions, by  M  of  the  prime  circle,  is  =  the  product  of  the  area  of 
the  circumscribing  circle  of  the  given  p,  by  A  of  a  like  p  to  the 
given  p. 

When  d  of  any  given  p  is  ^,  the  cube  of  the  area  of  such  p 
is  one  fourth  of  the  cube,  or  third  power,  of  M  of  a  like  p. 

So  the  third  or  cube  root  of  the  square  of  M  of  any  given  p 
is  m  of  a  like  jo,  when  the  area  is  the  third  root  of  such  M. 

The  area  of  any  given  p  is  to  the  square  of  d  of  such  j9,  as  M 
of  a  like  p  is  to  1. 

So  the  area  of  any  given  p  is  to  the  square  of  m  of  such  p  as 
d  when  m  is  1.,  is  to  1. 

When  the  area  of  a  JP  is  1.,  the  product  of  d  by  b  is  =  b  of  a 
like  jo,  whose  m  is  1. 

Such,  then,  are  a  few  of  the  facts,  corollaries,  and  conse- 
quents, flowing  from  that  vast  store-house  of  demonstrated  prin- 
ciples, which  the  world  has  accumulated. 

And  would  the  prescribed  limits  of  the  work  permit,  much 


54  ON    THE    QUADRATURE 

curious  evidence  might  be  deduced,  in  favor  of  my  determination 
of  the  quadrature,  from  a  consideration  of  the  sides  of  polygons, 
(which  I  denote  by  V,)  in  which  the  sum  of  the  squares  of  all 
the  several  sides  of  a  regular  polygon,  is  a  mean  between  the 
square  of  one  side  and  the  square  of  the  entire  perimeter. 
Thus,  when  V  of  the  p  4  is  1.,  the  area  is  1.,  or  =  ^4  when  w4 
is  1.  When  V  8  is  1.,  the  area  is  =  four  times  d  8  when  m  8  is 
1. ;  and  when  V16  is  1.,  the  area  is  =  sixteen  times  ^16  when 
m  16  is  1.,  &c. 

In  such  investigation,  as  also  in  a  consideration  of  the  sum  of 
all  the  powers  of  certain  polygonal  quantities,  we  should  dis- 
cover the  inclination  of  the  measures  of  the  p  8  to  be  governed 
by  equal  means,  in  lieu  of  proportional  means.  Thus,  the  sum 
of  all  the  powers  of  the  square  of  C  8  is  =  d 8  when  mS  is  1. ; 
the  sum  of  all  the  powers  of  A  8  is  twice  d  8  when  m  8  is  1. ;  so 
the  sum  of  all  the  powers  of  M  8  is  four  times  d  8  when  m  8  is 
1.,  or  four  times  the  reciprocal  of  M  8,  &c.  Nevertheless,  the 
rugged  octagon,  when  unfolded,  presents  great  mathematical 
beauties ;  and  number  8,  amongst  numerical  quantities,  is  cer- 
tainly a  conspicuous  number,  in  its  constant  application  in  the 
development  of  physical  laws  by  the  aid  of  numbers.  I  will 
here  suggest,  (before  proceeding  further  with  explanatory  mat- 
ter,) a  simple  polygonal  progression,  based  upon  the  principle, 
that  any  given  point,  either  above  or  below  ^  is  half  the  square 
of  "4^  above  a  mean  between  such  point  and  ^ ;  and  hence,  the 
progression  will  be  alike  applicable  to  all  polygons,  from  the  jt?3 
to  the  circle,  and  is  designed  to  approximate  I)  and  B  to  a  coin- 
cidence. 

B  of  any  given  p  is  half  the  square  of  T  above  &  of  a  like  ;?, 
whose  area  is  T  below  the  adverse  of  B  of  the  given  p ;  and  D 
of  any  given  p  is  half  the  square  of  T  above  d  of  a  like  jo, 
whose  area  is  ^  below  the  adverse  of  D  of  the  given  p. 

^  is  a  mean  between  B  4  and  ^,  and  B  4  is  half  the  square  of 
^  above  ^  ;  and  when  b  4  is  ^  the  area  of  such  p4=  is  T  below 
the  adverse  of  B  4. 

So  D  4  is  ^,  and  is  half  the  square  of  T  above  a  mean  be- 
tween D  4  and  ^  ;  and  when  d  4  is  a  mean  between  D  4  and 
^,  the  area  of  such  p  4  is  ^  below  i,  or  is  T  below  the  adverse 
ofD4. 

Hence,  such  progression,  by  an  infinite  series,  will  extinguish 
the  mean  between  B  and  ^,  upon  the  one  hand,  and  between  D 
and  £,  upon  the  other ;  nor  can  the  progression  proceed  further. 
And  it  is  manifest  that,  at  the  end  of  the  progression,  either  in 
respect  to  B  or  D,  that  the  area  will  be  .5,  or  f  below  B  or  D, 
namely,  T  below  ^. 


OF    THE    CIRCLE.  55 

From  the  fact,  that  the  product  of  the  perimeter  of  any  regu- 
lar jf?,  by  one  fourth  of  the  diameter  of  the  inscribed  circle  of 
such  polygon,  gives  the  area  of  such  p,  it  follows  as  a  corollary, 
that  if  the  square  root  of  such  perimeter  be  taken  as  the  hypoth- 
enuse  of  a  right  angled  triangle  of  two  equal  legs,  or  sides,  that 
the  square  of  the  distance  from  the  centre  of  the  base  to  the 
right  angle,  (namely,  the  square  of  the  altitude,)  will  be  =  the 
area  of  the  given  p.  Let  such  perimeter,  then,  be  the  perimeter 
of  a  major  polygon ;  in  which  case,  the  square  of  the  altitude 
will  always  be  =  M  of  the  given  p. 

In  such  case,  the  base  will  always  be  =  m  of  a  like  j9,  whose 
area  is  4 ;  one  of  the  equal  sides  will  be  =  m  of  a  like  />, 
whose  area  is  2  ;  and  the  altitude  will  be  =  m  of  a  like  p, 
whose  area  is  1.  ;  and  the  square  of  the  altitude  will  be  =  M 
of  a  like  p;  one  of  the  equal  sides  will  be  =  b  of  a  p  4,  of  equal 
area  with  that  of  the  given  7?,  or  will  be  a  mean  between  2.  and 
M  of  the  given  p ;  and  when  M  of  the  given  p  is  ^,  b  of  an 
equal  p  4  will  then  be  the  reciprocal  of  ^. 

1  is  used  to  denote  the  difference  between  M  8  and  B  8,  or 
between  C  8  and  D  ;  which  difference  is  extinguished,  or  made 
JL  in  the  circle.  If,  then,  A  be  a  diminishing  quantity  from  the  p  8 
to  the  circle,  it  ought  to  be  so  denoted  in  the  course  of  the  pro- 
gression ;  thus,  if  in  the  p  16  it  shall  be  but  a  proportional  half, 
for  instance,  of  what  it  is  in  the  p  8,  and  so  on,  the  same 
should  appear,  as  thus,  i,  J,  J,  &c. ;  for  if  the  diminishing  quan- 
tity be  constantly  denoted  by  ^  we  shall  be  inclined  to  conceive 
the  same  difference  to  exist  between  M  and  B  or  C  and  D, 
when  the  same  is  wholly  extinguished,  as  existed  in  the  p  8. 

And,  in  fact,  when  M  and  C  become  adverse,  although  they 
then  are  in  actual  coincidence,  we  cannot  well  get  rid  of  the 
conception,  that  M  is  J.  above  ^  and  C  JL  below  ^ ;  for  as  we 
still  retain  in  our  conceptions  the  order  of  the  progression,  we 
still  conceive  the  terms  as  being  separate,  and  their  distance  as 
being  the  same  as  previously. 

SECTION    FOURTH. 

It  will  not,  probably,  be  doubted  or  denied,  that  the  order  in 
which  the  four  prime  denotations  progress,  from  the  p  4  to  the 
prime  circle,  or  to  the  coincidence  of  C  and  M  and  the  coinci- 
dence of  D  and  B,  is  as  follows,  namely,  C  :  D  : :  M :  B,  or 
C  :  M  : :  D :  B  ;  for  such  is  the  popular  progression  and  final 
determination.  The  difference  between  M  and  B  and  between 
C  and  D  being  £  in  the  p  4,  but  greater  in  the  p  8  than  in  the 
p  6. 


56  ON    THE    QUADRATURE 

Neither  will  it  be  denied  that,  by  the  popular  method,  the  act- 
ual difference  between  M  and  B  and  between  C  and  D,  is  in- 
creased from  the  p  4  to  the  circle  ;  so  that  the  difference  is  even 
greater  (saying  nothing  about  ratio,)  in  the  circle  than  it  is  in 
the  p  8. 

Neither  will  it  be  denied,  that  the  progressive  ratios  of  M  and 
D  may  coincide  in  ^,  and  that  they  cannot,  upon  any  system  or 
principle,  coincide  in  any  other  point  or  position ;  and  that  the 
same  will  be  the  case  in  respect  to  A  and  B,  or  C,  and  B  of  twice 
the  sides,  in  case  they  are  permitted  to  progress  thus  far ;  for 
such  is  manifestly  the  law  of  progression,  by  any  system  or 
method. 

So  it  is  manifest,  also,  that  by  the  general  law  of  progression, 
when  the  progressive  ratios  of  M  and  D  coincide  in  ^,  if  C  and 
B  do  not  then  coincide  with  M  and  D,  they  will  be  adverse ; 
so  also,  if  M  and  C  of  a  p  of  half  the  sides  of  that  whose  M 
and  D  are  in  ^/,  are  not  also  in  ,£,,  they  must  necessarily  be 
adverse. 

Hence,  when  M  and  D  coincide  in  ^  if  a  like  or  a  similar  dif- 
ference still  prevails  between  M  and  B  and  between  C  and  D 
as  in  the  p  8,  the  formula  for  the  four  prime  denotations,  for  the 
p  whose  M  and  D  are  in  ^  and  the  p  of  half  the  sides,  will  be 
thus :  C  :  D  C  :  D  M  :  B  :  M  :  B.  But  if,  when  M  and  D  coin- 
cide in  3>  me  difference  between  M  and  B  and  between  C  and 
D  is  JL,  or  is  equated  to  JL,  then  all  the  above  eight  denotations 
are  in  ^ ;  or,  if  we  choose  so  to  consider  it,  those  denotations 
belonging  to  the  p  whose  M  and  D  apparently  coincide  in  ^ 
are  extinguished,  by  the  coincidence  of  those  of  half  the  sides, 
which  may  be  shown  to  be  universally  the  case  in  respect  to  all 
denotations  (of  whatever  kind)  which  apparently  coincide  in 
^;  and  hence,  we  may  either  consider  them  as  vanishing,  or 
becoming  extinguished  by  like  denotations  of  half  the  sides,  in 
the  point  of  apparent  coincidence ;  or  we  may  consider  them  as 
still  existing  in  coincidence  with  those  of  half  the  sides. 

And  to  show  that  such  phenomena  must  take  place,  (confining 
our  investigations  wholly  to  major,  minor  and  primal  polygons,) 
one  side  of  the  minor  p  4,  when  properly  equated  and  adjusted, 
is  as  well  the  base  of  two  adjoining  sides  of  the  major  p  8,  as 
of  two  adjoining  sides  of  the  minor  p  8;  and  the  like  is  true,  in 
respect  to  the  p  8  and  p  16,  and  so  on,  ad  infinitum,  until  one  side 
of  a  minor  p  is  =  two  sides  of  either  a  major  or  minor  p  of 
twice  the  sides ;  or,  rather,  until  all  numeral  difference  between 
them  is  extinguished. 

The  application,  thus  far,  in  cases  in  which  one  and  two  sides 
have  been  considered,  has  been  general,  and  is  adapted  to  any 


OF    THE    CIRCLE.  57 

series  of  polygons;  but,  in  order  to  avail  ourselves  of  the  full 
benefits  of  the  phenomena,  we  must  confine  ourselves  to  that 
series  of  polygons  which  commences  with  the  p  4  ;  for  the 
reason,  that  in  such  a  series  M  or  C  of  any  /?,  (except  that  of  the 
p  4,)  is  composed  of  an  even  number  of  sides ;  and  hence,  from 
the  p  8  onward  to  the  circle,  the  phenomena  apply  equally  as 
well  to  M  or  to  C,  as  to  one  or  two  sides,  which  is  not  the  case 
with  any  other  series  than  that  commencing  with  the  p  4. 
Hence,  the  commencement  of  our  process  is  properly  with  the 
p  8 ;  and  especially,  when  we  wish  to  show  the  progress  of  a 
polygonal  progression  downward,  towards  M  of  the  prime 
circle. 

Thus  C  of  the  p  8  may  be  said  to  span  both  C  and  also  M  of 
the  p  16,  and  so  on,  until  C  becomes  =  C,  as  also  M,  of  a  p  of 
twice  the  sides,  which  can  only  occur  in  the  circle ;  and  the  con- 
sequence necessarily  is,  that  in  either  JP,  C  is  =  M,  which  may 
be  called  the  fourfold  chord  of  the  periphery. 

Hence  M,  in  descending,  must  necessarily  become  =  M, 
as  also  =  C  of  a  p  of  twice  the  sides ;  and  the  only  ques- 
tion is,  in  what  numerical  point  or  position  shall  this  fourfold 
chord  be  formed  or  situated  ?  If  it  be  in  ^  tnen  B  and  D  final 
are  in  ^ ;  if  it  be  above  ^,  then  D  and  B  final  will  be  below  %, ; 
and  if  it  be  below  ^,  then  D  and  B  final  will  be  above  3  j  f°r 
if  it  be  not  in  ^,  then  ^  will  De  tne  farther  of  two  means  from 
the  coincidence  of  M  and  C  to  that  of  B  and  D. 

If  M  and  C  coincide  in  ^,  the  difference  between  M  and  B 
and  between  C  and  D  will  have  become  equated  to  _2_.  If  M 
and  C  coincide  below  ,£,,  such  difference  will  remain,  or  has 
been  increased  or  diminished  from  what  it  was  in  the  p  8,  ac- 
cordingly as  it  shall  be  in  the  circle.  And  if  M  and  C  shall 
coincide  above  ^,  then  such  difference  will  have  been  extin- 
guished, and  will  have  again  increased  in  the  opposite  direction. 
And  certainly  I  will  not  contend  in  favor  of  the  last  position,  but 
it  is  my  business  to  contend  against  the  second. 

This  fact,  alone,  then,  would  seem  to  present  an  insurmount- 
able difficulty  in  the  way  of  the  popular  mode,  namely,  when 
the  progressive  ratios  of  M  and  D  coincide  in  ^ ;  if  we  attempt 
to  extend  the  progression  farther,  M  and  B  will  in  future  pro- 
gress by  the  same  ratio,  and  C  and  D  will  also  progress  by  the 
same  ratio,  which  is  contrary  to  the  popular  law  of  progression, 
in  which  the  difference  between  M  and  B,  as  also  between  C 
and  D  is  constantly  increasing  from  the  p  8  to  the  circle.  And 
certainly  the  very  law  of  progression  itself,  ought  not  to  be  re- 
quired to  give  way,  under  any  circumstances.  Besides,  M  and 
B  are  properly  regarded  as  descending  quantities  ;  and  two 
8 


58  ON   THE    QUADRATURE 

separate  quantities,  descending  by  the  same  ratio,  must  necessa- 
rily converge. 

Hence,  it  is  impossible  that  the  progression  can  be  extended 
after  M  and  D  coincide,  without  an  alteration  of  the  law  of  pro- 
gression. If,  then,  the  law  of  progression  is  not  altered  in  ^ 
then  M  and  C,  and  B  and  D  coincide  in  ^. 

We  perceive,  that  C  and  C  and  M  of  a  p  of  twice  the  sides, 
(in  the  series  commencing  with  the  p  4,)  preserve  their  relative 
situations  from  the  p  8  and  p  16  to  the  circle ;  and  D  of  the  p 
whose  M  and  C  are  adverse,  and  D  and  B  of  the  p  of  twice  the 
sides,  (namely,  the  p  whose  M  and  D  are  in  ^,)  have  the  same 
relative  situations,  in  respect  to  each  other,  as  C  of  a  p  whose 
M  and  C  are  adverse,  and  C  and  M  of  the  p  of  twice  the  sides, 
have. 

I  have  said,  that  from  the  p  8  to  the  circle  the  same  relative 
situation,  between  C,  and  C  and  M  of  a.p  of  twice  the  sides,  is 
constantly  preserved ;  and  it  may  be  of  consequence  to  know, 
whether  from  the  p  8  to  the  circle  the  same  relative  situation  is 
constantly  preserved  between  C  and  M  of  a  p  and  C  and  M  of 
ap  of  twice  the  sides. 

A  mean  between  C  and  M  is  to  a  mean  between  D  and  B,  as 
C  is  to  D  or  as  M  is  to  B ;  or  a  mean  between  M  8  and  C  8  is 
_1  below  a  mean  between  B  8  and  D  8 ;  hence,  if  M  16  be  a 
mean  between  M  8  and  C  8,  it  is  JL  below  a  mean  between  B  8 
and  D  8. 

Now  M  8  is  a  mean  between  M  4  and  %_  below  D  4,  and  M  8 
is  also  a  mean  between  C  4  and  JL  below  B  4 ;  and  if  M  16  be 
a  mean  between  M  8  and  C  8,  then  the  quantity  denoted  by  Ji 
must  be  constantly  halved.  Now  M  16  is  to  a  mean  between 
M  8  and  C  8  as  it  is  to  a  mean  between  C  8  and  M  8,  for  they 
are  one  point,  and  so  on.  So  M 16  is  to  a  mean  between  M  8 
and  JL,  below  D  8,  as  it  is  to  a  mean  between  C  8  and  J_  below 
B  8,  for  they  are  one  and  the  same  point. 

If,  then,  M  16  is  a  mean  between  M  8  and  JL  below  D  8  or 
C  8,  and  JL  below  B  8,  we  then  have  the  terms  of  a  progression 
from  the  p\  to  the  circle,  in  which,  from  the  pS  to  the  circle,  the 
same  relative  position  will  constantly  hold  between  M  and  C  of 
any  p  and  M  and  C  of  a  p  of  twice  the  sides ;  and  it  would 
seem,  in  reality,  as  if  such  progression  was  corroborated  by  the 
popular  determination. 

If,  then,  the  p  4  be  called  the  primitive  p  of  the  series,  and  all 
the  other  polygons  of  the  series  be  called  secondary  polygons, 
then,  while  C  16  is  a  mean  between  M  16  aad  C  8,  M  16  will 
be  a  mean  between  M  8  and  C  8,  and  so  on  to  the  circle. 

But  to  proceed  with  the  progression  on  the  foregoing ;  M  8  is 


OF    THE    CIRCLE.  59 

a  mean  between  M  4  and  A  below  D  4,  and  between  C  4  and 
2_  below  B  4,  (for  they  are  one  and  the  same.)  M  16  is  a  mean  be- 
tween M  8,  and  !_•  below  D  8,  and  between  C  8  and  J_  below  B  8. 
M  32  is  a  mean  between  M  16,  and  £  below  D  16,  and  between 
C  16,  and  J  below  B  16,  and  so  on. 

Such  being  the  case  it  is  manifest  that  all  the  four  denotations 
must  coincide  together,  which  must  of  necessity  be  in  ^.  Or, 
on  the  same  principle,  we  have  this  progression  :  A  8  is  to  M  8 
as  M  8  is  to  L  below  B  4  ;  A  16  is  to  M  16,  as  M  16  is  to  1  be- 
low B  8,  and  so  on.  Or  thus ;  M  4  is  to  M  8,  as  M  8  is  to  A  be- 
low D  4,  M  8  is  to  M  16,  as  M  16  is  to  .1  below  D  8,  and  so  on. 

Thus  A  and  M  would  coincide  with  B  of  half  the  sides,  and 
M  and  D  would  coincide  with  M  of  twice  the  sides.  And  this 
entirely  comports  with  the  law  of  progression.  So  again,  C  4  is 
to  M  8,  as  M  8  is  to  1  below  B  4 ;  C  8  is  to  M  16,  as  M  16  is  to 
_1  below  B  8,  and  so  on. 

But  the  popular  methods  instead  of  equating  JL  to  JL,  have 
only  equated  it  to  Ji,  and  that  by  placing  M  and  C  final,  f  be- 
low their  proper  place. 

But  a  great  number  of  progressions  of  the  kind,  and  on  simi- 
lar principles,  may  be  instituted  —  all  of  which  will  find  their 
termini  in  ^ :  for  m  of  any  given  p  when  the  area  is  .5,  namely, 
M  by  D,  or  C  by  B,  of  any  given  p  -r-  by  ^  gives  m  of  a  like 
p,  when  the  area  is  ^. 

M  of  any  given  jo,  is  to  b  when  the  area  is  1,  as  C  is  to  d 
when  the  area  is  1,  and  the  proportional  difference  between  M 
4,  and  b  4,  when  the  area  is  1,  is  less  in  the  p  4,  than  in  any  p 
between  the  p  4  and  circle  ;  but  this  proportional  difference  is  the 
same  in  the  p  4  and  the  circle.  Hence  the  linear  capacities  are 
the  same  in  the  p  4  as  in  the  circle.  The  difference  between  M 
and  6,  when  the  area  is  1,  or  between  C  and  d,  when  the  area  is 
1,  is  greater  in  the  p  8  than  in  any  p  of  the  series  commencing 
with  the  p  4,  and  is  denoted  by  1_ ;  hence  the  p  8  is  the  proper 
p  from  which  to  commence  our  equation  of  polygonal  capacities, 
whether  it  be  mathematically  or  geometrically  performed. 

M  8  is  J_  below  a  mean  between  M  4  and  I)  4,  or  C  4  and  B  4, 
and  M  16  is  J  below  a  mean  between  M  8  and  D  8,  or  C  8  and  B 
8,  and  so  on.  And  by  such  progression,  it  is  manifest  that  all 
the  terms  coincide  in  ^. 

Now  no  one  will  deny  but  that  M  8  is  _L  below  the  square 
root  of  m  4,  when  the  area  of  such  p  4  is  .5 ;  nevertheless,  the 
popular  method  has  placed  M  of  the  prime  circle,  J  below  the 
square  root  of  m  of  the  circle  whose  area  is  .5,  from  which  it  is 
manifest  that  by  such  method,  no  progress  has  been  made  from 
the  p  16  to  the  circle,  but  that  there  has  even  been  a  retrograde 


60  ON    THE    QUADRATURE 

from  the  truth,  occasioned  by  an  increase  of  the  constant  divisor 
in  an  incorrect  proportion  to  that  of  the  constant  dividend. 

I  have  said  it  might  be  important  to  know  avhether  from  the 
p  8  to  the  circle,  M  and  C,  and  M  and  C  of  a  p  of  twice  the 
sides,  constantly  retained  the  same  relative  position  in  respect  to 
each  other.  It  is  quite  certain  that  they  do  not  from  the  p  4  to 
the  circle.  Then  draw  the  circle  whose  diameter  is  assumed  at 
unity )  and  from  any  point  in  the  periphery,  draw  one  side  of  the 
perimeter  of  the  major/?  8,  (namely,  one  side  of  M  8,)  as  a  tan- 
gent, and  such  tangent  may  be  said  to  be  the  tangent  to  one  side 
of  C  8,  and  also  to  two  sides  of  M  16,  as  also  two  sides  of 
C  16,  one  side  of  M  16  being  merged  in  such  tangent,  which 
side  of  M  16  so  merged,  is  the  tangent  of  two  sides  of  M  32,  of 
two  sides  of  C  32,  and  of  one  side  of  C  16,  —  and  so  on,  ad  in- 
ftnitum,  until  the  tangent  is  extinguished,  namely,  until  M  and 
C  coincide  with  M  and  C  of  twice  the  sides,  or  until  M  and  C 
have  extinguished  M  and  C  of  twice  the  sides,  thus  determining 
the  fact,  that  from  the  p  8  to  the  circle,  M  and  C  and  M  and  C 
of  twice  the  sides  constantly  retain  the  same  relative  situation  to 
each  other.  And  no  fact  can  be  made  more  manifest,  than  that 
when  M  is  in  ^  M  and  C  of  half  the  sides  are  adverse,  and  D  of 
the  p  whose  M  is  in  *,  is  =  M ;  C  of  such  p  is  a  mean  between 
M  of  such  p,  and  C  of  half  the  sides.  B  of  such  p  is  a  mean 
between  M  or  D  of  such  p,  and  M  of  half  the  sides. 

What  then  must  be  the  relative  situation  of  M  16  in  respect 
to  M  8  and  C  8  ?  Let  legitimate  geometry  answer  the  question. 

Hence,  M  16  is  a  mean  between  C  8  and  J_  below  B  8,  or 
is  a  mean  between  M  8  and  _L  below  D  8.  Or  M  16  is  J  below 
a  mean  between  M  8  and  D  8 — or  J  below  a  mean  between  C 
8  and  B  8,  So  M  32  is  J  below  a  mean  between  M  16  and  D 
16,  or  C  16  and  B  16,  and  so  on,  ad  infinitum,  until  M  finally 
coincides  with  M  and  D  of  half  the  sides  and  with  C  and  B  of 
half  the  sides.  And  as  before  shown,  such  coincidence  is  in  ,£,. 

We  hence  obtain  or  avail  ourselves  of  the  true  equation  of 
polygonal  measures  ;  but  to  attempt  this  without  either  unity  of 
purpose,  or  legitimate  geometry,  would  be  a  hopeless  task. 

Hence  if  any  final  denotation  is  in  ^,  they  all  become  final 
in  *, ;  in  which  case  also,  if  any  denotation  appears  to  be  in  ^ 
it  is  there  extinguished  by  a  like  denotation  of  half  the  sides. 
And  it  might  be  a  pleasant  exercise  in  mathematics,  to  multiply 
auxiliary  denotations  of  other  polygonal  quantities ;  as  for  in- 
stance, a  mean  between  C  and  D,  which  is  also  a  mean  between 
A  and  B  —  and  a  mean  between  M  and  D,  which  is  also  a  mean 
between  C  and  B,  and  is  the  square  root  of  m  of  a  like  p  whose 
area  is  5,  or  half  the  square  of  the  reciprocal  of  A,  or  of  C,  or 
of  M  and  C ;  all  of  which,  by  the  laws  of  polygonal  progression, 


OF    THE    CIRCLE.  61 

are  seen  to  progress,  or  flow  into,  and  to  coincide  in  ,£,.  But  I 
forbear  to  enter  upon  it  further.  It  may,  however,  be  suggested, 
that  if  we  signify  ihe  p  whose  M  and  D  appear  to  coincide  in  ^, 
by  placing  figure  2  at  the  right  of  its  denotations,  and  figure  1  at 
the  right  of  the  denotations  of  the  p  whose  M  and  C  appear  to  be 
adverse,  then  when  M  2  and  D  2  appear  to  coincide  in  £,,  C  1 
and  M  1  actually  coincide  in  ^,  in  which  case  M  1  will  appear 
to  assume  the  identical  place  of  C  1,  when  M  1  and  C  1  are 
conceived  to  be  adverse  ;  and  if  the  progression  be  conceived  to 
be  still  continued  (for  we  may  conceive  a  continuance  of  the 
progression  until  we  can  conceive  quantity  to  be  extinguished 
by  constantly  halving  it)  M  and  B  will  thence  be  conceived  to 
ascend,  and  C  and  D  to  descend,  contrary  to  the  order  or  law  of 
polygonal  progression,  which  process,  or  imaginary  process,  (for 
it  is  but  imaginary,)  will  present  many  curious  coincidences  and 
adverse  reciprocities  by  way  of  the  reductio  ad  absurdum,  which 
I  will  not  here  enter  upon. 

But  I  will  cease  to  multiply  evidence  of  this  description,  as- 
suring the  world  that  the  store  house  is  yet  full,  and  by  no  means 
exhausted  by  the  pittance  which  I  have  drawn  from  it. 

Z  may  denote  a  mean  between  M  and  D  or  C  and  B  of  any 
given  p,  which  is  at  the  same  time  the  square  root  of  m  of  a  like 
p  whose  area  is  .5.  Then,  from  the  p  4  to  the  circle,  Z  and  M 
of  a  p  of  twice  the  sides,  approximate  each  other,  by  any  system. 
Then,  Z  4  is  1  above  M  8,  Z  8  is  £  above  M 16,  Z  16  is  ±  above  M 
32,  and  so  on,  till  Z  is  =  M  of  twice  the  sides. 

Z  is  the  farther  of  three  means  from  M  to  £,.  Hence,  Z  can- 
not be  =  M  of  twice  the  sides  below  ,£,,  for  in  that  case  M  would 
be  below  M  of  twice  the  sides,  which  is  impossible. 

If  from  the  p  8  to  the  circle,  a  mean  between  M  and  C  is  M 
of  twice  the  sides,  a  progression  on  this  principle  will  be  identical 
with  that  by  which  we  say  that  Z  4  is  _1  above  M  8,  Z  8  is  £  above 
M  16,  &c.  And  then,  from  the  p  8  onward,  D  is  to  a  mean 
between  D  and  C,  as  a  mean  between  M  and  I)  is  to  M  of  twice 
the  sides,  or  as  Z  is  to  M  of  twice  the  sides,  or  as  Z  is  to  a  mean 
between  M  and  C  of  the  same  p,  which  last  also  applies  to  the 
1*4 

Again,  when  Z  becomes  a  mean  between  M  and  C  (=  M  of 
twice  the  sides)  such  mean  between  M  and  C  (or  M  of  twice  the 
sides)  is  a  mean  between  M  and  D,  or  C  and  B,  or  B  and  D. 
Then  M  is  =  B,  and  C  is  =  D ;  and  M  of  twice  the  sides  is  also 
=  B  of  such  p  of  twice  the  sides :  thus  in  two  intermediate  poly- 
gons, M  is  =  B,  and  when  M  is  =  B,  C  is  =D. 

Because  M8  is  a  mean  between  M4  and  j±  below  D  4,  it  is 
a  mean  between  C  4  and  JL  below  B  4.  Because  M  16  is  a 


62  ON    THE    QUADRATURE 

mean  between  M  8  and  J_  below  D  8,  it  is  a  mean  between  C  8 
and  Ji  below  B  8,  and  so  on. 

So  when  M  becomes  a  mean  between  M  and  D,  or  C  and  B 
of  half  the  sides,  it  is  also  a  mean  between  M  and  C  of  half  the 
sides.  When  M  is  given,  we  conceive  M  of  twice  the  sides, 
even  though  there  be  no  difference  between  them. 

Hence,  when  M  and  D  coincide,  we  conceive  M  of  twice  the 
sides,  even  though  it  be  =  such  M  in  ^  as  a  mean  between  M 
and  D  in  £,. 

In  order. that  M  may  be  a  mean  between  M  and  D  of  half  the 
sides,  it  must  be  the  farther  of  three  means  from  M  of  half  the 
sides,  to  %,. 

M  below  ^,  cannot  be  a  mean  between  M  and  D  of  half  the 
sides,  for  in  that  case,  M  of  half  the  sides  would  be  the  lower 
M,  for  M  cannot  be  above  M  of  half  the  sides. 

By  no  system  is  M  ever  above  a  mean  between  M  and  D  of 
half  the  sides,  that  is,  a  mean  between  M  and  C  is  never  above 
a  mean  between  M  and  D. 

Whenever  M  is  a  mean  between  M  and  D  of  half  the  sides, 
it  is  also  a  mean  between  M  and  C  of  half  the  sides,  in  which 
case  C  and  D  are  necessarily  =.  So  when  there  is  no  difference 
between  M  and  B,  or  C  and  D,  a  mean  between  M  and  C  of 
such  p  is  M  of  twice  the  sides. 

By  the  popular  method,  M  does  not  become  a  mean  between 
M  and  C  of  half  the  sides,  except  in  the  circle,  and  hence,  M  is 
always,  (from  M  16  to  the  circle,)  £  below  a  mean  between  M 
and  I),  or  C  and  B.  But  the  fact  is  too  manifest  to  be  doubted, 
that  M  in  ^  is  a  mean  between  M  and  C  of  half  the  sides. 

When  M  is  in  £,,  Z  is  necessarily  in  ^,  being  then  the  farther 
of  three  means  from  M  to  %,. 

Z  is  above  M  of  twice  the  sides,  until  they  become  =. 

In  ^  M  is  =  Z  of  the  same  p.  Below  ^,  M  cannot  be  =  Z 
of  any  p.  Hence  M  cannot  be  =  Z  of  half  the  sides,  except 
when  Z  and  M  of  the  same  p  are  =  and  such  must  be  the  case 
in  the  point  in  which  M  coincides  with  M  of  twice  the  sides. 
Such  point  is  in  £,. 

The  progression  may  also  be  carried  on  thus  :  The  square  root 
of  m  4,  when  the  area  is  .5,  is  J_  above  M  8.  The  square  root  of 
m  8,  when  the  area  is  .5,  is  £  above  M  16,  &c.,  until  the  square 
root  of  m,  when  the  area  is  .5,  is  =  M  of  twice  the  sides. 

Now  it  is  manifest  that  the  square  root  of  w,  when  the  area  is 
.5,  cannot  be  =  M  of  twice  the  sides,  in  any  point  above  ,£,,  for 
it  is  only  in  ^  that  it  becomes  =  M  of  a  like  p,  for  a  mean  between 
M  and  D,  (which  is  the  square  root  of  m  when  the  area  is  .5,) 
is  above  until  M  and  D  coincide  in  . 


OF    THE    CIRCLE.  63 

It  is  equally  manifest,  that  the  square  root  of  m  when  the  area 
is  .5,  cannot  be  =  M  of  twice  the  sides  below  ,£,,  for  the  reason 
that  the  square  root  of  m,  when  the  area  is  .5,  is  always  the  far- 
ther of  three  means  from  M  to  ^ ;  hence,  for  the  square  root  of 
m,  when  the  area  is  .5,  to  be  =  M  of  twice  the  sides,  in  any 
point  below  ^,  would  require  M  to  be  below  M  of  twice  the 
sides,  which  would  be  a  phenomenon  that  no  one  would  expect. 

Hence  it  is  determined  in  the  most  conclusive  manner,  that  ,£, 
is  the  point  in  which  we  must  conceive  the  square  root  of  m, 
when  the  area  is  .5,  to  become  =  M  of  twice  the  sides ;  while 
in  the  same  point,  it  is  =  M  of  the  same  p. 

Hence,  ^  is  the  very  point  in  which  M  apparently  becomes 
=  M  of  twice  the  sides,  which  is  a  phenomenon  that  apparently 
must  occur  in  the  formation  of  the  periphery  of  the  circle.  Nev- 
ertheless, such  M  of  twice  the  sides  does  not  actually  exist,  it 
being  but  the  conception  of  M  of  twice  the  sides  of  that  M  which 
has  no  sides,  or  the  number  of  which  (if  number  it  may  be  called) 
is  infinite.  And  should  any  require  corroborative  evidence  in 
favor  of  my  determination  that  M  of  the  prime  circle  is  in  ^, 
let  them  apply  to  legitimate  geometry. 

The  equation  of  the  capacities  of  polygonal  measures,  properly 
commences  with  the  p  8,  in  which  p  there  is  the  greatest  differ- 
ence between  M  and  B,  or  C  and  D,  and  this  is  plainly  dictated 
by  geometry.  And  in  reference  to  the  varying  capacities  of  po- 
lygonal measures,  the  jo4,  or  unit  polygon,  is  the  perfect  standard, 
the  true  place  of  any  one  of  its  measures  always  being  in  the 
mean  place.  But  not  so  in  respect  to  polygons  situated  between 
the  p  4  and  its  inscribed  circle ;  in  which  the  linear  capacities  are 
variable,  the  true  places  varying  from  the  mean,  and  hence  requir- 
ing a  constant  equation,  the  variance  being  greater  in  the  p  8  than 
in  any  other  p. 

Thus  the  variance  of  the  true  place  of  m  8  from  the  mean  place, 
is  proportionally  twice  as  great  in  respect  to  d8,  as  it  is  in  respect 
to  b  8,  and  d  8  is  proportionally  as  much  above  its  mean  place,  as 
m  8  is  below,  while  b  8  is  always  in  its  mean  place  in  respect  to 
area,  as  well  in  the  p  8  as  in  all  polygons  from  the  p  4  to  the  circle ; 
for  it  must  be  recollected  that  these  varying  capacities  have  regard 
or  reference,  to  the  admeasurment  of  area. 

Nevertheless,  the  linear  capacities  of  the  circle  are  the  same  as 
those  of  the  p  4 ;  and  hence,  in  a  progression  from  the  p  4  to  the 
circle,  those  varying  capacities  must  be  regarded  in  order  to  avoid 
constant  and  final  error. 

This  must  be  done  by  geometrizing  wholly  by  means  of  lines, 
(which  is  the  only  legitimate  geometry,)  without  any  regard 
whatever  to  areas,  in  the  course  of  our  progression ;  for  we  may 


64  ON    THE    QUADRATURE 

be  assured,  that  when  our  lines  are  properly  adjusted  and  pro- 
portioned, that  the  areas  will  be  so  too.  But  it  is  too  manifest, 
that  if  we  attempt  to  progress  by  means  of  areas  instead  of  lines, 
that  we  shall  wholly  fail  to  equate  the  varying  linear  capacities  of 
polygons  situated  between  the  j»4  and  its  inscribed  circle. 

I  have  said  that  if  one  final  denotation  be  in  ^,  (however  the 
denotations  be  multiplied,)  then  all  the  denotations  will  necessa- 
rily coincide  together  in  ^ ;  but  if  any  final  denotation  be  not  in 
^,  then  the  final  denotations  will  be  scattered,  some  being  above 
%,,  and  some  below,  but  none  in  £,. 

And  for  the  purpose  of  exhibiting  the  phenomenon,  I  will  still 
increase  the  number  of  denotations  expressed  by  capital  letters. 

E  may  denote  half  the  square  of  the  reciprocal  of  A,  of  any 
given  p.  F  may  denote  half  the  square  of  the  reciprocal  of  C 
of  any  given  p;  namely,  half  the  square  of  b  of  a  p  whose 
m  is  1,  —  and  G  may  denote  half  the  square  of  the  reciprocal  of 
M;  namely,  half  the  square  of  d  of  a  p  whose  m  is  1.  Now 
half  the  square  of  the  reciprocal  of  C  of  any  given  p  is  =  the 
area  of  a  like  p  whose  m  is  =  B  of  a  like  p ;  and  in  the  p  8 
such  area,  or  half  the  square  of  the  reciprocal  of  C  8,  is  an  equal 
mean  between  M  4  and  C  4,  and  is  the  proper  divisor  for  ob- 
taining M  8,  when  the  dividend  is  the  product  of  M  4  by  C  4, 
and  by  such,  or  a  similar  divisor  does  the  popular  method  essay 
to  carry  on  a  progression  ;  but  it  will  not  succeed  beyond  obtain- 
ing M  8.  So  also,  half  the  square  of  the  reciprocal  of  M  is  = 
the  area  of  a  like  p  when  m  is  =  D  of  a  like  p. 

B  then  of  any  p  is  a  mean  between  M  and  half  the  square  of 
the  reciprocal  of  C  of  a  like  p,  and  D  of  any  p  is  a  mean  be- 
tween M  and  half  the  square  of  the  reciprocal  of  M  of  such  p. 

B  of  any  given  p  is  the  farther  of  three  means  from  half  the 
square  of  the  reciprocal  of  A  of  a  like  p  to  ^  —  and  D  of  any 
given  p  is  the  farther  of  three  means  from  half  the  square  of  the 
reciprocal  of  M  to  ,£,. 

Hence,  a  mean  between  half  the  square  of  the  reciprocal  of  C 
and  ^,  is  the  adverse  of  C,  and  a  mean  between  half  the  square 
of  the  reciprocal  of  M  and  ^,  is  the  adverse  of  M. 

But  I  will  not  further  multiply  terms,  as  any  one  can  multiply 
them  at  pleasure,  and  discover  a  vast  variety  of  polygonal  coin- 
cidences and  reciprocities,  very  many  of  which  I  have  examined, 
but  have  not  here  even  hinted  at. 

Let  then  any  denotation  which  I  have  signified  by  a  capital 
letter  be  final  in  ^. 

Let  A  then,  be  considered  final  in  ^  j  tnen  B  is  in  ,£,,  for  A 
and  B  coincide  in  ^ ;  then  C  of  half  the  sides  is  final  in  £, ; 
so  M  is  final  in  the  point  in  which  C  is  final ;  D  coincides  with 


OF    THE    CIRCLE.  65 

M  of  a  like  p  in  ^ ;  so  then  D  is  final  in  ^ —  Z  is  final  in  ^, 
because  it  is  a  mean  between  M  and  D,  or  C  and  B.  A  being 
in  ,2^,  hence  E  is  in  ^,  for  I)  is  half  the  square  of  the  reciprocal 
of  A  ;  and  the  like  may  be  said  in  respect  to  C  and  F,  or  of  M 
and  G ;  —  or  thus  —  if  d  a  denote  the  point  in  which  d  and  the 
area  of  any  given  p  are  =,  which  is  always  the  reciprocal  of  M, 
or  is  d  when  m  is  1 ; — if  b  a  denote  the  point  in  which  b  and  the 
area  of  any  given  p  are  =,  which  is  the  reciprocal  of  A  of  a 
like  p,  then  D  is  the  farther  of  three  means  from  half  the  square 
of  d  a  to  ^ ;  hence  when  D  is  in  ^,  half  the  square  of  d  a  is  in 
2, ;  when  A  and  B  are  in  ^,  half  the  square  of  b  a  is  in  £, ;  and 
when  C  is  in  ^,  half  the  square  of  b  a  of  twice  the  sides  is  in  2,. 
Hence,  if  any  denotation  be  final  in  ^,  there  is  then  a  general 
coincidence  of  all  the  special  denotations  in  ^  excepting  d  a  and 
b  a,  which  are  then  reciprocals  of  ^,  and  is  the  diameter  of  an 
orbit  in  which  the  period  is  .5.  I  might  have  here  added  another 
very  important  denotation,  signifying  a  mean  between  C  and  D, 
(as  Z  does  between  M  and  D,)  which  denotation  may  be  ex- 
pressed by  Y,  which  is  always  the  farther  of  three  means  from  A 
to  ,£,,  and  the  square  of  which  is  a  mean  between  .5  and  A,  of  a 
like  p ;  hence,  when  A  is  in  ^,  Y  is  in  ^,  at  the  same  time 
such  Y  being  a  mean  between  C  and  D,  which  are  also  in  ^; 
so  Z  in  ^  is  a  mean  between  M  and  D  when  they  coincide  in  %,. 

Hence  ^  seems  to  be  emphatically  the  goal  of  the  spirit  land, 
and  deserves  far  more  consideration  than  the  narrow  limits  which 
I  have  assigned  to  this  work  will  justify  ;  it  being  my  design  to 
select  a  few  of  the  dictums  and  corollaries  from  amongst  those 
which  I  have  examined  and  considered,  and  those  too  which  are 
least  complex,  or  may  be  most  easily  understood. 

But  in  respect  to  that  focal  point  denoted  by  ^,  at  which  all 
polygonal  denotations  centre  and  find  their  home,  it  is  possible 
that  a  little  farther  hovering  around  it,  by  way  of  "  closer  and 
closer  contact,"  may  not  be  inappropriate  at  the  close  of  this 
chapter. 

Now  if  the  difference  between  M  and  B  or  C  and  D  be  con 
stantly  denoted  by  _L,  (as  would  be  the  case  by  the  popular  meth- 
od, save  that  JL  becomes  enlarged  from  the  p  8  to  the  circle,  by 
the  popular  mode,)  there  will  be  no  difficulty  in  determining  the 
relative  situation  of  the  denotations  of  any  /?,  whenever  any  two 
denotations  coincide,  or  those  of  a  p  of  half  the  sides,  or  of 
twice  the  sides,  &c.  For  in  such  case,  conception  will  always 
place  B  1  above  M,  and  D  1  above  C.  So  F  will  be  1  above 
M,  and  G  will  be  JL  above  A,  by  conception ;  and  such  is  the 
conception  in  the  finale,  or  in  the  circle.  Nevertheless,  the  popu- 
lar determination  furnishes  as  ample  proof  as  can  be  desired 


66  ON    THE    QUADRATURE 

that  the  progressive  ratios  of  M  and  D,  of  A  and  B,  of  C  and 
B  of  twice  the  sides,  of  D  and  Z,  of  B  and  Y,  &c.  &c.,  have  their 
point  of  coincidence  in  ,£,. 

Let  M  and  D  then  coincide  in  ^,  and  we  conceive  B  to  be  L 
above  M,  and  D  JL  above  C  ;  hence,  when  M  and  D  are  in  £,  B 
and  C  are  adverse.  But  as  C  is  =  A  of  twice  the  sides,  hence, 
C  of  the  p  whose  M  is  in  ^,  is  a  mean  between  ^,  and  C  of 
half  the  sides ;  that  is,  C  of  half  the  sides  will  be  conceived  to 
be  J.  below  ^,  and  as  D  is  J_  above  C,  hence,  C  of  the  p  whose 
M  is  in  ^  is  =  D  of  a  p  of  half  the  sides,  (and  here  I  will 
make  the  allegation,  that  D  can  never  be  above  C  of  twice  the 
sides,  nor  can  M  be  below  B  of  twice  the  sides.) 

Thus  we  find  D  to  be  1  below  ^  —  and  as  tL  ™  tne  farther 
of  two  means  from  M  to  D,  hence  M  1  is  conceived  to  be  1. 
above  ^,  or  the  adverse  of  C  1. 

Hence,  M  and  D  of  the  p  2  is  in  ^,  and  B  and  C  of  the  same 
p  are  adverse,  and  M  and  C  of  the  p  1,  are  adverse ;  and  hence  it 
is  easy  to  assign  all  the  relative  denotations  of  these  two  poly- 
gons. 

In  such  case,  as  we  still  conceive  the  difference  between  B  and 
M,  or  D  and  C,  to  be  the  same  as  in  the  p  8,  (namely,  denoted 
by  J_)  we  necessarily  conceive  M  and  C  of  the  p  2  to  be  sepa- 
rated by  the  same  amount  as  M  and  B,  or  C  and  D. 

Hence,  if  we  suppose  such  difference  to  be  reality,  in  lieu  of 
mere  conception,  then  the  progression  must  be  carried  on,  ad  in- 
finitum ;  and  it  is  quite  manifest  that  such  fancied  progression 
would  have  a  termini  nearly  (though  not  quite)  in  the  two  points 
of  the  popular  termini ;  —  not  quite,  I  say,  because,  from  the  co- 
incidence of  M  and  D  in  ^,  the  popular  law  of  progression  must 
be  altered  from  what  it  has  previously  been,  as  is  quite  manifest, 
for  after  M  and  D  coincide,  C  and  D  must  necessarily  thereafter 
progress  by  the  same  ratio ;  so  also  must  M  and  B,  &LC.  ;  and  a 
mean  between  M  and  C  thenceforward  will  be  M  of  twice  the 
sides ;  besides,  if  M  and  D  descend  by  the  same  ratio,  they  will 
converge  towards  each  other,  and  hence,  on  this  account  alone, 
will  place  the  conceived  termini  of  the  final  M  above  the  popu- 
lar place.  This  is  a  portion  of  the  subject  that  may  be  exempli- 
fied to  great  extent,  but  I  forbear. 

Y  is  the  constant  mean  between  C  and  D,  and  is  the  farther 
of  three  means  from  A  to  ^ ;  hence,  when  Y  is  in  ^,  A  is  also 
in  ^,  and  consequently,  B  is  in  %,. 

Z  is  the  constant  mean  between  M  and  D,  and  between  C  and 
B,  and  when  M  and  D  are  in  ^,  Z  is  a  mean  between  them ; 
or  when  Z  is  in  %,f  M  and  D,  as  also  the  area  of  such  major  p 
are  in  %,.  Thus  when  Z  is  in  ^,  it  is  =  M,  or  the  area  of  such 
major;?;  and  is  =  D  of  a  like;?. 


OF    THE    CIRCLE.  67 

So  when  Y  is  in  £,,  it  is  =  A  and  B  of  such  jo,  for  Y  is  the 
farther  of  three  means  from  A  to  ^ ;  and  A  is  =  C  of  half  the 
sides  until  A  and  C  become  =,  namely,  until  they  become  =  M. 

Hence  A  in  ^  is  =  C,  for  Y  cannot  be  =  C  of  half  the  sides ; 
and  when  Y  is  =  C,  it  is  also  =  D,  for  Y  is  a  mean  between  C 
and  D.  Hence  C  in  £,  is  =  M. 

We  shall  all  agree  that  if  the  polygonal  denotations  expressed 
by  capital  letters,  be  considered  in  the  nature  of  flowing  quantities, 
from  the^4,  or  rather  from  the  pS,  preserving  the  same  relative 
situation  to  each  other,  (viz.  those  descending  and  also  those  as- 
cending,) until  they  commence  forming  coincidences  in  ^,  that 
Y  and  Z  will  coincide  at  the  same  time  (if  not  at  the  same 
place)  that  M  and  C  do. 

We  shall  all  agree  that  the  square  of  Z  is  always  a  mean  be- 
tween .5  and  M,  and  that  the  square  of  Y  is  always  a  mean 
between  .5  and  A ;  consequently,  when  the  square  of  Y  becomes 
a  mean  between  .5  and  C,  C  is  =  M  ;  and  if  such  be  in  ^,  then 
all  the  denotations  will  coincide  together  in  ^.  Hence  the  place 
of  the  coincidence  of  Y  and  Z  is  important.  Y  commences 
from  A  of  the  p  8 ;  and  Z,  from  B  of  the  p  8,  and  the  popular 
conclusion  is,  that  they  coincide  £  below  ^,  that  A,  C  and  M 
coincide  §  below  ^,  that  B  and  D  coincide  £  above  ^  and  E,  F 
and  G  |  above  ^.  So  the  popular  conclusion  is,  that  the  deno- 
tations on  their  passage  to  their  final  coincidences,  often  salute 
each  other  and  pass  on ;  thus  M,  Z,  D  and  G  coincide  in  ^,  but 
each  is  destined  to  pass  on  in  order  to  form  other  coincidences. 
So  also  M,  and  B  of  twice  the  sides,  (according  to  the  popular 
method,  will  coincide  f  above  ^,  or  the  adverse  of  the  popular 
coincidence  of  M  and  C,  with  others  formed,  and  again  dispersed 
in  the  vicinity  of  ^.  But,  by  the  popular  mode,  not  a  single 
final  coincidence  is  in  £,.  Nevertheless,  if  Y  and  Z  were  to  co- 
incide in  ^,  all  the  other  denotations  would  coincide  in  £,.  Hence, 
if  either  denotation  be  final  in  ^,  all  the  denotations  would 
coincide  in  ^  J  in  which  case  the  capacities  of  the  linear  meas- 
ures of  the  circle  will  be  the  same  as  those  of  the  p4,  and  that 
they  are  so,  is  certainly  sustained  by  too  much  evidence,  both 
general  and  special,  to  be  doubted  or  denied. 

Our  principal  object  is,  to  ascertain  the  point  in  which  C 
shall  become  =  A,  or  when  C  shall  become  —  M,  viz.  when  all 
surrounding  area  shall  be  excluded  from  A,  of  the  circle,  by  the 
periphery ;  or  perhaps,  after  the  manner  of  Leibnitz,  that  nu- 
meral position  in  which  the  curve  of  the  circle  shall  osculate  all 
the  angles  contained  as  well  in  the  perimeter  of  the  circumscribed, 
as  of  the  inscribed  polygon.  That  is,  when  a  mean  between  Y 
and  Z,  (which,  as  in  other  cases,  is  always  Y  of  twice  the  sides,) 


DO  ON    THE    QUADRATURE 

becomes  extinguished  by  the  coincidence  of  Y  and  Z,  the  square 
of  either  Y  or  Z  will  be  a  mean  between  .5  and  the  osculating 
point,  or  point  in  which  C  becomes  =  M,  or  is  merged  in  M.  If 
then  Y  final  be  in  ^,  its  square  is  a  mean  between  .5  and  A, 
which  is  then  =  M ;  its  square  is  also  a  mean  between  .5  and  C, 
which  is  then  =  M. 

A  polygonal  progression,  then,  does  not  (as  has  been  sup- 
posed,) depend  upon  an  alternate  succession  of  equal  and  rational 
means  ;  but,  on  the  contrary,  from  the  p  8  onward  to  the  circle, 
namely,  throughout  the  whole  progression  through  the  secondary 
polygons  of  the  series,  it  proceeds  wholly  by  rational  means; 
in  which,  what  may  be  called  a  denotation  of  twice  the  sides  of 
its  fellows  is  always  a  mean  between  them.  Thus,  b  a  8  is  a 
mean  between  b  a  4  and  d  a  4 ;  b  a  16  is  a  mean  between  b  a  8 
and  d  a  8,  and  so  on,  ad  infinitum.  So  b  a  8  is  =  b  4,  when  m  4 
is  1,  and  so  on  to  a  coincidence  in  the  third  or  cube  root  of  2. 
So  a  mean  between  E  and  G  is  E  of  twice  the  sides,  unto  a 
coincidence  of  E  and  G ;  a  mean  between  B  and  1)  is  B  of  twice 
the  sides,  ad  infinitum,  and  the  like  is  the  case  in  respect  to  Y 
and  Z. 

Such  being  the  premises,  the  conclusion  need  not  be  mis- 
taken. 

As  a  kind  of  synopsis  to  the  quadrature,  (upon  which  too 
much  evidence  cannot  well  be  adduced,)  it  may  not  be  improper 
to  offer  it  in  a  dress  or  version  somewhat  new. 

Thus  b  a  may  denote  the  point  in  which  b  and  the  area  of 
any  given  p  are  =;  da  may  denote  the  point  in  which  d  and 
the  area  of  a  p  are  = ;  M  a  may  denote  the  point  in  which  m 
and  the  area  of  a  p  are  = ;  (which  point  has  heretofore  been 
denoted  by  M.)  C  denotes  a  mean  between  M  a  and  A. 

A  mean  between  b  a  and  da  of  any  given  p  is  =  b  a  of  a 
p  of  twice  the  number  of  sides ;  hence,  a  mean  between  b  a  and 
da  of  any  given  j9,  is  the  reciprocal  of  a  mean  between  A  and 
M  a  of  a  like  p. 

B  is  the  farther  of  three  means  from  E,  or  half  the  square  of 
b  a  to  2,,  and  D  is  the  farther  of  three  means  from  G,  or  half 
the  square  of  d  a  to  %,  j  Y  is  the  farther  of  three  means  from  A 
to  3,  and  is  a  mean  between  C  and  D ;  and  Z  is  the  farther 
of  three  means  from  M  a  to  ^,  and  is  a  mean  between  M  a 
and  D. 

Thus,  a  mean  between  d  a  and  b  a  is  b  a  of  a  p  of  twice  the 
sides,  until  such  mean  between  d  a  and  b  a  no  longer  exists,  or 
until  b  a  is  merged  and  extinguished  in  da  of  the  circle, 
namely,  until  there  is  no  longer  da  or  b  a  of  a  p  of  sides. 

When  d  a  and  b  a  coincide,  it  is  manifest  that  A  and  C  be- 


OF    THE    CIRCLE,  69 

come  merged  and  extinguished  in  Ma  of  the  circle ;  for  A  is 
the  reciprocal  of  b  a,  and  C  is  the  reciprocal  of  a  mean  be- 
tween b  a  and  d  a.  And  the  point  in  which  b  a  becomes  merged 
in  D  «,  is  the  reciprocal  of  the  point  in  which  A  and  C  become 
merged  in  M  a  by  any  system. 

A  mean  between  E  and  G  is  =  E  of  twice  the  sides,  until 
E  becomes  merged  in  G.  A  mean  between  B  and  D  is  =  B 
of  twice  the  sides,  until  B  is  merged  in  D.  And  a  mean 
between  Y  and  Z  is  =  Y  of  twice  the  sides,  until  Y  is  merged 
in  Z. 

Hence,  we  may  very  properly  consider,  that  on  arriving  at  the 
quantities  and  measures  of  the  circle,  all  means  between  similar 
denotations  are  extinguished,  or  do  not  exist.  Hence,  that  in  the 
quantities  or  measures  of  the  circle,  there  is  neither  b  a,  E,  B, 
A,  C,  or  Y,  as  all  those  (while  they  exist)  denote  polygons  of 
sides. 

Nevertheless,  in  the  final  event,  the  only  proper  quantities  and 
measures  of  the  circle,  for  which  we  are  seeking,  are  those  ex- 
pressed by  M  a  and  D ;  and  the  progressive  ratios  of  M  a  and  D 
are  found,  by  any  system,  to  coincide  in  ^. 

The  object  of  research,  however,  is  to  ascertain  the  point  in 
which  b  a  becomes  extinguished  in  d  a,  for  in  the  reciprocal  of 
such  point  A  becomes  merged  or  extinguished  in  Ma;  and  in  a 
like  p,  namely,  in  the  circle,  B  becomes  extinguished  in  D.  Y  in 
Z,  and  E  in  G. 

So  long  as  b  a  exists,  we  necessarily  conceive  of  a  mean  be- 
tween b  a  and  d  a ;  and  the  like  is  the  case  in  respect  to  E  and 
G,  B  and  D,  A  and  M  a,  or  Y  and  Z.  Hence,  we  may  as  well 
conceive  that  d  a  coincides  with  b  a  of  twice  the  sides,  as  with 
b  a  of  a  like  number  of  sides  with  da;  for  d  a  forms  a  coin- 
cidence as  much  with  one  as  with  the  other.  Hence,  if  we  say 
that  da  coincides  with  b  a  of  twice  the  sides,  it  is  also  a  coin- 
cidence with  b  a  of  the  same  number  of  sides  with  da,  (if  we 
may  speak  of  number  where  there  is  no  number,)  and  the 
like  is  true  in  respect  to  D  and  B,  of  Ma  and  A,  of  G  and  E, 
or  Z  and  Y. 

Now  A  8  is  JL  below  half  the  square  of  d  a  8,  and  M  a  8  is 
JL  below  half  the  square  ofba  of  twice  the  sides ;  and  such  is, 
by  the  popular  determination,  conceived  to  be  the  relative  situa- 
tion of  A  and  M  a,  onward  to  the  circle,  in  which  case  such  JL  is 
supposed  constantly  to  increase  in  value. 

We  have  retained,  then,  as  measures  or  quantities  of  the 
circle,  (after  the  extinguishment  of  E,  B,  C,  A  and  Y,)  those 
of  Ma,  D,  Z  and  G ;  and,  by  the  popular  method  of  considering 
the  subject,  as  well  as  by  any  other  method  that  may  be  de- 


70  ON    THE    QUADRATURE 

vised,  those  four  proper  measures,  or  quantities,  of  the  circle, 
coincide  together  in  £,.  That  is,  all  the  proper  measures  of 
the  circle,  situated  below  unity,  necessarily  coincide  together 
in  ^;  hence,  if  either  proper  measure  of  the  circle  is  in  ^ 
the  others  are  also. 

But  again,  C,  or  a  mean  between  A  and  M  a,  coincides  with 
B  of  twice  the  sides  of  such  Ma;  and  we  also  say  that  D  coin- 
cides with  B  of  twice  the  sides.  But,  by  the  popular  method, 
one  of  these  coincidences  would  be  in  ^,  while  the  other  would 
be  i  above  ^.  So  M  a,  D,  Z  and  G,  (the  proper  measures  of 
the  circle,)  coincide  in  ^,  and  A,  B,  Y  and  E,  (extinguishable 
quantities,)  are  conceived  to  coincide  together  in  ^.  But  A,  B, 
Y  and  E  are  as  well  means  as  principals  ;  and  hence,  M  a,  D,  Z 
and  G  will  coincide  in  ^ ;  A,  B,  Y  and  E  will  coincide  in  ^ ; 
and  a  mean  between  A  and  M  a,  a  mean  between  B  and  D,  a 
mean  between  E  and  G,  and  a  mean  between  Y  and  Z,  coin- 
cide together  in  £,;  and  this  upon  the  simple  principles  of  po- 
lygonal progression,  which  all  may  readily  examine,  and  of  the 
truth  of  which  all  may  satisfy  themselves ;  and  such  phenomena 
cannot  exist,  unless  ^  be  the  point  in  which  the  proper  measures 
of  the  circle  coincide,  and  in  which  all  other  polygonal  quanti- 
ties cease  to  exist. 

Nevertheless,  when  we  consider  ^  as  the  point  in  which  a 
mean  between  A  and  M  a,  between  B  and  D,  between  E  and  G, 
and  between  Y  and  Z  coincide,  we  necessarily,  in  our  concep- 
tion, transpose  the  terms  of  half  the  sides,  and  imagine  those 
which  have  previously  been  above  %,  to  nave  passed  below  ^, 
and  those  which  have  been  previously  below  ^  to  nave  passed 
above  ^ ;  which  necessarily  arises  from  our  conception,  that 
there  is  a  progression  so  long  as  we  retain  the  terms  denoting 
polygons  of  sides. 

Thus,  if  we  conceive  such  means  to  coincide  in  ^  that  is,  if 
we  conceive  C,  as  a  mean  between  A  and  M  a,  to  be  in  ^,  Ma 
will  be  conceived  to  occupy  the  identical  place  that  A  does, 
when  Ma  is  in  ^ ;  and  A  will  appear  to  occupy  the  place  of 
Ma  of  half  the  sides  of  Ma  in  ^  ;  B  will  be  in  the  place  of  C, 
when  M  a  is  in  ^ ;  and  D  will  be  in  the  place  of  B,  when  Ma 
is  in  ^,  &c.  But  this  is  but  conception,  founded  upon  that 
which  still  conceives  the  progression  to  be  continued ;  which, 
if  true,  would  place  M  a  below  A  or  C,  B  below  D,  E  below 
G,  and  Z  below  Y  ;  and  such  a  phenomena  no  one  expects 
to  see. 

When  Ma  and  D  are  in  ^, such  Ma  or  D  is  half  the  square 
of  da;  and  when  A  and  B  are  in  3>  such  A  or  B  will  be  half 
the  square  of  b  a. 


OP    THE    CIRCLE.  71 

And  if  we  conceive  C  to  be  in  ^,  namely,  a  mean  between 
A  and  M  a,  such  C  will  be  conceived  to  be  half  the  square  of  a 
mean  between  d  a  and  b  a ;  in  which  case,  such  d  a  will  be 
conceived  to  be  above  b  a,  being  an  event  which  cannot  occur. 

Let  us  suppose  ^  to  be  the  termini  of  an  infinite  series,  for 
the  progressive  ratios  of  M  a  and  D,  and  of  A  and  B;  B  4  and 
M«4  are  in  1,  D  4  in  ^,  and  A 4  in  ^  If  we  progress,  by 
successive  means,  between  M  a  4  and  ,£,,  and  between  D  4  and 
,2,,  the  termini  will  be  in  ^;  or  if  we  proceed,  by  successive 
means,  between  B  4  and  £,  and  between  A  4  and  ^,  the  termini 
will  also  be  in  %,.  But  in  such  case,  we  conceive  the  last  step 
in  the  progression,  — from  D4  to  ^, —  to  be  one  step  in  ad- 
vance of  a  like  number  of  steps,  in  the  progression  from  A  4  to 
3-  That  is,  in  the  coincidence  of  D  with  ^,  we  conceive  A  to 
be  one  step  behind  or  below  ^,  which  step  we  conceive  to  be 
JL  below  ^.  And  hence  we  conceive,  that  when  D  (and  conse- 
quently M  a)  is  in  ^,  B  of  the  same  number  of  sides  is  J_  above 
£,  ;  being  thus  transported,  by  our  conception,  from  the  fact, 
that  in  proceeding  from  M  a  4,  or  from  B  4,  we  have  progressed 
or  travelled  in  the  same,  or  in  the  identical  footsteps  in  both 
cases,  and  also  by  the  same  number  of  footsteps.  And  in  such 
progression,  —  from  M  a  4  to  ^and  from  D  4  to  ^,  or  from  B  4 
to  £,  and  from  A  4  to  ^,  —  the  results  will  be  the  same,  whether 
we  progress  by  a  constant  mean  or  by  the  farther  of  two,  the 
farther  of  three,  and  so  on,  ad  infinitum,  until  numbers  or  means 
are  exhausted,  or  to  an  actual  coincidence.  Nevertheless,  the 
most  natural  method,  and  at  the  same  time  that  affording  the  most 
ready  explanation  and  evidence,  is  probably  that  proceeding  by 
the  farther  of  two  means. 

In  such  case,  (as  by  any  other  progression,)  when  the  progres- 
sive ratios  of  M  a  and  D  coincide  in  ,£,,  we  conceive  A  to  be  £ 
below  „£,,  and  B  J_  above  ^  5  nevertheless,  we  have  passed  or 
progressed  by  the  same  steps,  in  respect  to  B  as  to  M  a,  from 
their  coincidence  in  the  p  4,  and  hence  the  farther  of  two  means 
arrives  as  soon  in  ^  in  one  case  as  in  the  other. 

And  to  aid  us  in  a  proper  conception  of  the  matter,  the  econ- 
omy of  numbers,  and  that  of  quantities,  are  so  curiously  adjusted 
to  each  other,  that  we  need  not  remain  in  much  doubt ;  for  if  we 
conceive  another  step  of  the  progression  in  respect  to  A  and  B, 
we  shall  find  A  and  B  in  their  vanishing  points  according  to  the 
popular  determination,  —  that  is,  in  the  points  in  which  they 
become  merged  or  extinguished  in  M  a  and  D  of  the  circle ;  for 
by  such  additional  step,  we  conceive  A  to  be  f  below  ^,  and  B 
i  above  £,  or  in  the  popular  M  a  and  D  of  the  circle. 

The  utmost  then,  which  can  be  claimed,  is,  that  when  M  a 


72  ON    THE    QUADRATURE 

and  D  coincide,  the  very  next  step  in  the  progression,  applied  to 
A  and  B,  extinguishes  them.  And  the  same  remark  applies  to 
Y  and  to  E,  for  by  such  progression,  when  M  a  and  D  are  in  ^, 
(and  consequently  Z  and  G,)  the  very  next  step  in  the  progres- 
sion, applied  to  Y  or  to  E,  places  them  in  Z  and  G  final,  accord- 
ing to  the  popular  notion  of  their  final  positions. 

And  because  G  and  Z  are  in  „£,  when  M  a  and  D  are  contained 
in  it,  and  because  E  and  Y  coincide  with  A  and  B  in  ,£,,  and 
are  extinguished  in  the  same  p  as  are  A  and  B  in  our  present 
consideration  of  the  subject,  we  may  dispense  with  G  and  Z,  as 
also  with  E  and  Y,  and  merely  consider  C  as  a  mean  between 
A  and  M  a.  Thus  in  the  foregoing  example  of  a  progression 
from  M  a  or  B  of  the  p  4,  or  from  A  and  from  D  of  the  p  4,  if 
we  adopt  a  simple  mean  in  lieu  of  the  farther  of  two  means, 
then  when  M  a  and  D  are  in  ^  if  we  conceive  A  and  B  to 
take  another  step  in  the  progression,  we  shall  conceive  it  to  place 
A  JL  below  ^j  and  B  ^  above  „£,.  But  if,  when  M  a  is  in  3/>  the 
quantity  denoted  by  i_  is  already  extinguished,  except  in  the 
imagination  or  conception,  then  A  and  B  are  in  ^  when  M  a 
and  D  are ;  in  which  case,  such  A  and  B  of  twice  the  sides  will 
only  be  in  their  respective  vanishing  means. 

When  either  B  or  D  coincide  with  a  mean  between  B  and  D, 
it  is  but  the  coincidence  of  B  and  D,  or  rather  the  extinguish- 
ment of  B  as  well  as  that  of  a  mean  between  B  and  D ;  notwith- 
standing we  may  still  conceive  both  B  and  such  mean  to  exist; 
and  the  like  is  true  in  respect  to  A,  and  M  a,  &c.,  in  which  case, 
even  such  extinguished  mean  will  be  conceived  to  be  B  or  A  of 
twice  the  sides ;  for  from  the  p  8  onward,  B  is  but  a  mean  between 
B  and  D  of  half  the  sides,  and  A  is  but  a  mean  between  A  and 
M  a  of  half  the  sides ;  and  when  those  means  are  extinguished, 
M  a  and  D  only  remain  as  proper  quantities  of  the  circle. 
Hence,  as  C  is  but  a  mean  between  A  and  M  a,  when  such  mean 
is  extinguished,  C  is  also  extinguished,  or  is  merged  in  M  a. 

Thus  the  extinguishment  of  A  and  B  is  the  extinguishment 
of  flowing  polygonal  quantities,  or  rather  of  quantities  denoting 
polygons  of  sides ;  and  we  must  conceive  them  to  flow  until 
those  quantities  which  necessarily  denote  polygons  of  sides 
are  extinguished,  and  until  we  can  have  no  conception  of  a 
farther  progression,  or  until  any  farther  attempt  at  a  progression 
will  apparently  cause  a  retrograde  movement. 

Now  the  progressive  ratios  of  A  and  B  coincide  in  ^,  by  any 
system,  and  if  ^  be  the  point  of  their  extinguishment,  it  is  then 
the  point  of  a  general  coincidence  ;  and  it  is  also,  clear,  that  we 
cannot  conceive  an  extension  of  the  progression  beyond  the 
coincidence  of  A  and  B,  for  an  attempt  at  a  farther  progression 


OP    THE    CIRCLE.  73 

gives  an  apparent  retrograde  movement,  and  transposition  of 
terms. 

Thus,  if  A  and  B  coincide  in  ^,  we  conceive  M  a  of  a  like 
p  to  be  L  below  £,  (going  upon  the  ground  that  from  the  p  8,' 
or  even  from  the  p  4,  M  a  shall  be  constantly  denoted  by  J_  be- 
low B.)  But  A  in  ^,  as  in  any  other  case,  is  a  mean  between 
A  and  M  a  of  half  the  sides ;  or,  as  has  all  along  been  consider- 
ed, is  =  C  of  half  the  sides. 

C  in  ^,  then,  denotes  a  p  of  half  the  sides  of  that  whose  A  is  in 
,2,,  and  C  is  a  mean  between  A  and  M  a  of  the  p  which  it  de- 
notes, and  C  in  ^  denotes  a  p  whose  M  a  is  A  below  ^  and 
whose  A  is  A  above  ^,  (occupying  the  identical  places. of  Ma 
and  C  of  the  p  of  half  the  sides  of  that  whose  M  a  and  D  are 

in£.) 

Thus  where  A  and  B  coincide,  (or  appear  to,)  we  cannot  con- 
ceive of  a  p  of  twice  the  sides,  for  polygonal  quantities  denoting 
sides,  are  then  extinguished,  and  if  we  conceive  a  p  of  half  the 
sides,  (namely,  the  p  whose  C  is  conceived  to  be  in  ,£,,)  we  find 
A  to  be  JL  above  ,£,,  namely,  ^_  above  A  of  twice  the  sides ; 
which  phenomenon  will  not  be  required  by  any  one. 

Thus  the  final  or  extinguished  mean  between  A  and  M  a, 
and  the  final  or  extinguished  mean  between  B  and  D  coincide 
in  ^ ;  and  until  such  apparent  coincidence,  we  may  conceive  a 
continuance  of  the  progression. 

Then  considering  C  to  denote  simply  a  mean  between  A  and 
M  a,  when  C  is  conceived  to  be  in  ,£,,  it  is  but  the  point  of  its 
extinguishment ;  namely,  the  point  of  extinguishment  between 
A  and  M  a.  Hence,  in  a  general  consideration  of  the  quantity 
denoted  by  C,  its  more  proper  consideration  is  that  of  simply  a 
mean  between  A  and  M  a ;  nevertheless,  its  consequence  is 
vastly  important  when  considered  as  one  of  the  prime  linear 
quantities. 

Our  utmost  researches  then,  are  not  only  extended  to  extin- 
guished means  in  which  certain  approximating  quantities  coin- 
cide, but  also  to  the  general  coincidence  of  those  extinguished 
means ;  which  general  coincidence  is  in  ^,  by  any  rational  pro- 
gression that  may  be  devised.  And  then,  and  then  only,  do  we 
find  the  capacities  of  polygonal  measures  to  again  become  the 
same  as  they  were  in  the  p  4,  and  hence  to  be  capable  of  a  ra- 
tional application. 

I  have  before  remarked  upon  the  allegation,  which  we  often 
find  repeated  by  eminent  geometers  and  mathematicians,  that  a 
polygon  is,  properly  speaking,  the  area,  and  not  its  linear  meas- 
ures, as  is  often  supposed ;  but  I  have  been  rather  inclined  to 
adopt  the  vulgar,  or  common  acceptation,  believing  it  to  be 
10 


74  ON   THE    QUADRATURE 

best  suited  to  the  subject,  if  but  one  is  to  be  adopted.  Never- 
theless, with  a  view  to  a  full  consideration  of  the  subject,  the 
areas  should  be  considered  in  connection  with  their  linear  meas- 
ures ;  at  least  so  far  that,  whenever  linear  measures  of  polygons 
are  to  be  compared  and  proportioned  to  each  other,  each  linear 
measure  shall  denote  or  express  the  amount  of  area  of  the  poly- 
gon whose  linear  measure  is  given,  or  of  a  polygon  of  half  the 
sides,  or  of  twice  the  sides ;  thus  B  or  D  always  denotes  the  area 
to  be  .5 ;  M  a  or  M  denotes  either  the  area,  or  one-fourth  of  the 
circumference  of  a  major  polygon ;  while  C  denotes  one  fourth 
of  the  circumference  of  a  minor  polygon,  and  also  the  area  of  a 
minor  polygon  of  twice  the  sides. 

Hence,  by  the  use  of  M,  C,  B  and  D  we  have  as  full  and  as 
proper  an  expression  or  denotation  of  the  areas  of  the  major 
and  minor  polygons,  as  also  of  all  primal  polygons,  at  least  from 
the  octagon,  (or  polygon  from  which  our  equation  commences,) 
as  though  the  areas  themselves  were  expressed. 

But  the  converse  of  this  could  not  occur,  and  more  especially 
in  respect  to  diametric  quantities  of  polygons ;  for  each  polygon 
of  sides,  while  it  has  but  one  perimeter,  has  two  diametric  quan- 
tities, which  must  be  considered  in  our  investigations ;  hence, 
while  by  means  of  the  linear  measures  we  may  denote  the  quan- 
tities of  all  necessary  areas,  we  cannot,  by  means  of  the  areas, 
denote  all  the  necessary  linear  measures. 

And  to  show  that  a  direct  use  of  areas  may  be  dispensed 
with,  and  that  we  may  conduct  our  progression  wholly  by  linear 
quantities,  it  is  only  necessary  to  allege  the  fact,  that  a  mean  be- 
tween A  and  B  of  any  given  jo,  is  also  a  mean  between  C  and 
D  of  the  same  p ;  and  hence,  when  A  and  B  shall  coincide,  it 
will  be  as  well  in  the  mean  point  between  C  and  D,  as  between 
A  and  B.  So  also,  when  C  and  D  coincide,  it  must  be  in  the 
mean  point  between  A  and  B  of  the  same  p ;  and  hence  we 
may,  in  general,  adopt  C  and  D  in  our  progression,  to  the  exclu- 
sion of  A  and  B. 

So  a  mean  between  M  (or  M  a)  and  D  is  also  a  mean  be- 
tween C  and  B  of  the  same  p ;  hence,  when  M  and  D  coincide, 
it  as  well  in  the  mean  point  between  C  and  B,  as  between  M 
and  D ;  hence,  when  C  and  B  shall  coincide,  it  must,  of  neces- 
sity, be  in  the  mean  point  between  M  and  D ;  hence,  in  our  pro- 
gression we  may,  in  general,  adopt  M  and  D,  to  the  neglect  of  C 
and  B,  —  regarding,  however,  the  important  fact,  that  C  is  to  D 
of  any  given  p  as  M  is  to  B. 

Then  the  farther  of  three  means  from  M  to  ^  (which  is  a  mean 
between  M  and  D,)  will  be  the  mean  place  of  M  of  twice  the 
sides ;  and  the  farther  of  three  means  from  C  to  ^,  (which  is  a 


OF    THE    CIRCLE.  75 

mean  between  C  and  D  of  twice  the  sides,)  will  be  the  mean 
place  of  C,  or  of  D  of  twice  the  sides. 

Then,  when  M  is  in  ^,  it  will  be  =  the  mean  place  of  M  of 
twice  the  sides ;  and  when  C  is  in  ^,  it  will  be  =  the  mean 
place  of  C  of  twice  the  sides.  The  consequence  is,  that  when 
A  is  in  ^,  such  A  is  then  =  the  mean  place  of  C  of  the 
same  p. 

It  is,  then,  equally  well  to  speak  of  the  true  and  mean  places 
of  C  and  of  M,  as  of  A,  or  of  the  area  of  a  major  p;  never- 
theless, if  we  adopt  M  a  in  lieu  of  M,  we  then  make  the  ap- 
plication of  the  true  and  mean  places,  as  well  to  the  area  as 
to  M. 

But,  in  respect  to  minor  polygons,  C  is  =  A  of  twice  the 
sides ;  nevertheless,  the  farther  of  three  means  from  A  is  the 
mean  place  of  A  of  twice  the  sides,  and  the  farther  of  three 
means  from  C  is  the  mean  place  of  C  of  twice  the  sides,  until 
the  progression  is  ended. 

Thus  the  true  and  mean  place  of  A  8  is  the  same,  while  the 
true  place  of  M  a  8  is  j_  below  the  mean  place  ;  so  also,  the  true 
place  of  C  8,  or  of  A  16,  is  J  below  the  mean  place. 

Y,  then,  may  denote  the  mean  place  of  C  and  of  D,  and  Z 
may  denote  the  mean  place  of  Ma  of  twice  the  sides,  or  of  B 
of  twice  the  sides ;  thus  B  8  is  as  well  in  its  mean  place  as  in 
its  true  place  ;  and  such  construction,  in  respect  to  the  true  and 
mean  places  of  M  a,  B,  C  and  D  are  not  only  in  perfect  accord- 
ance with  the  law  of  polygonal  progression,  but  the  same  is 
even  corroborated  by  the  popular  method. 

Thus  C  is,  proportionally,  half  as  far  below  the  mean  place  of 
C  or  D,  as  Ma  is  below  the  mean  place  of  Ma  or  B  ;  and  C  is 
proportionally  as  far  too  low,  in  respect  to  the  mean  place  of  D, 
as  in  respect  to  the  mean  place  of  B. 

Now,  while  the  true  place  of  C  8  is  a  mean  between  the  true 
place  of  A  8  and  the  true  place  of  M  a  8,  so  the  mean  place  of  C  8 
is  a  mean  between  the  mean  place  of  A  8  and  the  mean  place  of 
MaS]  and  the  like  principle  holds  throughout  the  progression. 
Or  thus,  the  mean  place  of  C  is  a  mean  between  the  mean  place 
of  A  and  the  mean  place  of  M  a,  or  of  B. 

When  A  is  in  „£,,  such  A  is  in  the  mean  place  of  C  of  the 
same  p ;  hence  such  A  (as  the  mean  place  of  C)  is  a  mean  be- 
tween the  mean  place  of  such  A  and  the  mean  place  of  M  a, 
or  B. 

So  A  in  ^  is  a  mean  between  its  mean  place  and  the  mean 
place  of  M  a,  or  B ;  A  in  ^  is  in  the  mean  place  of  C,  or  of  D ; 
and  C  in  ^  is  in  the  mean  place  of  C,  or  of  D  of  twice  the  sides. 
And  the  like  is  true  of  B  in  ^  as  in  the  case  of  A ;  so  M  a  and 


76  ON    THE    QUADRATURE 

D  in  %,  are  in  the  mean  place  of  C  and  B  of  the  same  p,  or  of 
M  a  and  B  of  a  p  of  twice  the  sides. 

Thus  infinite  wisdom,  which  has  furnished  geometry  and  ma- 
thematics with  the  best  of  reasons,  but  which  never  intended 
that  we  should  survey  a  field  or  measure  and  determine  the  con- 
tents of  an  area,  without  taking  into  the  account  both  the  cir- 
cumferential and  diametric  measures  of  such  area,  has,  in  the 
present  case,  —  as  if  to  invite  us  to  quadrate  our  ways  at  once, 
and  not  to  be  eternally  attempting  to  equate  them  to  some  sup- 
posed, but  erroneous,  rule  or  law,  —  so  contrived  and  adapted 
the  evidence,  as  not  only  to  show  us  the  necessity  of  an  equation 
of  the  varying  capacities  of  polygonal  measures,  as  we  progress 
from  the  square  to  its  inscribed  circle,  but  also  to  dictate,  in 
divers  ways,  the  manner  of  conducting  the  same.  The  neces- 
sity of  this  equation  can  only  be  detected,  by  a  comparison  of 
the  perimetric  and  diametric  quantities  of  polygons  ;  nor  can  the 
equation  be  made  but  by  some  law  of  progression,  by  which 
those  quantities  are  constantly  compared  with  each  other. 

C  is  to  D  as  M  a  is  to  B ;  and  this  is  the  form  in  which  the 
progression  proceeds,  from  the  trigon  to  the  circle  ;  nevertheless, 
in  the  p  4  the  relative  situations  of  C  and  D  are  changed,  as  is 
also  the  case  in  respect  to  M  a  and  B ;  C  and  D  being  equal  in 
the  p  4,  and  M  a  and  B  being  equal  in  the  p  4.  But  the  p  8  pre- 
sents the  variance  between  C  and  D,  or  between  M  a  and  B, 
whichuis  denoted  by  _L,  and  which  requires  to  be  equated  to  .£. 
in  the  circle,  in  lieu  of  being  increased  in  value  as  we  progress 
from  the  p  8  to  the  circle,  as  is  the  case  by  the  popular  determi- 
nation. 

It  may  not,  however,  be  surprising,  that  the  mind  should  not 
readily  assent  to  an  equation  of  the  difference  existing  between 
M  a  and  B  (and  consequently  between  C  and  D,}  of  the  p  8  to 
JL,  on  arriving  at  ^,  but  rather  that  it  should  conceive  the  same 
ratio  to  exist,  as  we  progress  onward  from  the  p  8 ;  and  hence, 
when  the  progressive  ratio  of  M  a  is  in  ^,  that  the  same  ratio  ex- 
ists between  M  a  and  B  as  existed  in  the/*  8  ;  and  hence,  the  actual 
numerical  difference  between  M  a  and  B,  when  M  a  is  in  ^, 
would  be  but  very  little  less  than  it  is  in  the  p  8.  And  such 
would  be  the  case,  were  toe  capacities  of  the  linear  measures  of 
all  the  polygons  situated  between  the  p  8  and  circle,  the  same 
as  they  are  in  the  p  8 ;  but  not  so  if,  from  the  p  8  towards  the 
circle,  the  capacities  of  the  measures  of  polygons  are  again  ap- 
proximating to  what  they  were  in  the  p  4,  from  whence  we  com- 
menced our  progression. 

But  having  found  in  the  p  8  (or  in  the  first  step  in  the  progres- 
sion,) a  difference  between  M  a  and  B,  or  between  C  and  Dr 


OF    THE    CIRCLE.  77 

and  finding  the  law  to  be  universal,  that  M  a  is  to  B  as  C  is  to 
D,  and,  consequently,  that  C  is  to  M  a  as  D  is  to  B ;  the  mind 
which  is  imbued  with  the  least  tincture  of  ratio  and  proportion 
will  naturally  attach  to  any  sign,  denoting  such  difference  in  the 
p  8,  the  conception  that  it  is  a  sign  denoting  ratio  and  proportion, 
and  that,  as  such,  it  will  accompany  the  polygonal  progression, 
and  that  it  will  constantly  be  denoted  by  the  same  character  or 
sign  as  is  used  for  the  p  8,  which,  if  it  were  simply  to  denote 
ratio,  would  only  become  extinguished  in  0  ;  whereas,  by  a  pro- 
per equation  of  the  capacities  of  polygonal  quantities,  it  necessa- 
rily becomes  extinguished  in  the  circle. 

The  consequence  of  such  natural  or  rational  conception  of  the 
mind  is,  that  when  M  a  and  D  (which  are  in  advance  of  their 
mean  places  from  the  p  8  onward)  coincide  in  ^,  or  in  their 
mean  places,  B  is  still  conceived  to  be  J_  above  M  a,  and  C 
1_  below  D  ;  .thus  conceiving  the  difference  of  A  between  C  and 
B,  notwithstanding  they  actually  coincide  with  Ma  and  D 

itt& 

Such  conception,  however,  is  not  unprofitable,  as  it  enables  us 
to  give  (in  a  formula,  if  we  please)  the  relative  situations  of  the 
polygonal  quantities  and  measures,  not  only  of  the  p  whose  M  a 
and  D  coincide  in  ^,  but  also  of  the  p  of  half  the  sides,  or  of 
twice  sides  ;  for  it  will  be  recollected  that  the  circle  is  composed 
of  a  p  of  half  the  sides  and  a  p  of  twice  the  sides,  as  well  from 
the  minor  as  from  the  major  polygons ;  which  phenomenon  is 
the  result  of  a  demonstration,  the  truth  of  which  is  readily  per- 
ceived. 

It  may  not  be  inappropriate  here,  to  suggest  something  farther 
in  reference  to  the  measures  and  quantities  of  the  p  8,  and  of  the 
important  positions  which  they  hold  in  polygonal  economy. 
Thus,  the  true  places  of  A  and  of  B  of  the  p  8,  are  also  their 
mean  places,  and  A  8  is  the  square  of  B  8.  And  while  B  8  is 
a  mean  between  B  of  any  given  p,  and  the  square  root  of  A  of 
such  p ;  or  between  D  of  any  given  p  and  the  square  root  of  M  a 
of  such  j?,  A  8  is  a  mean  between  B  and  the  square  root  of  Y  of 
any  given  p,  viz.  between  B  of  any  given  p,  and  the  square  of 
the  mean  place  of  C  or  D  of  such  p>  or  of  the  mean  place  of  A 
of  a  p  of  twice  the  sides. 

Hence,  if  B  of  a  p  be  in  ^  A  8  is  the  mean  between  such  B 
and  the  square  of  the  mean  place  of  C  or  of  D,  or  of  A  of  twice 
the  sides. 

But  B  in  ^  is  =  A,  by  any  system,  as  also  by  the  law  of 
polygonal  progression,  and  is  a  mean  between  C  and  D  of  the 
same  p, —  or  is  in  the  mean  place  of  C  or  D  of  the  same  p, —  and 
is  consequently  in  the  mean  place  of  A  of  twice  the  sides  ;  but, 


78  ON   THE    QUADRATURE 

as  I  have  before  remarked,  the  place  of  A  of  twice  the  sides  is 
nowhere  to  be  found. 

So  we  say  that  A  in  £,  is  =  C  of  half  the  sides ;  consequent- 
ly, C  in  £,  is  =  the  mean  place  of  C  or  of  D  of  twice  the  sides, 
and  A  in  ^  is  =  the  mean  place  of  A  of  twice  the  sides.  And 
because  M  #  in  ^  is  =  Z,  hence,  M  a  in  ^  is  =  the  mean  place 
of  M  a,  or  B  of  twice  the  sides.  The  consequence  is,  that  if  we 
progress  upon  the  mean  place  of  M  a  and  B,  and  upon  the  mean 
place  of  C  and  D,  they  will  eventually  coincide  with  the  true 
places,  and  with  each  other.  As  the  mean  place  of  M  a  or  of  B 
is  a  mean  between  M  a  and  D,  or  between  C  and  B,  —  namely, 
the  farther  of  three  means  from  M  a  to  ^,  —  hence,  if  our  progres- 
sive steps  are  upon  the  successive  mean  places  of  M  a  or  B,  we 
can  only  arrive  in  £,  by  an  infinite  series ;  and  as  a  mean  be- 
tween M  a  and  D  is  also  a  mean  between  C  and  B,  perhaps  no 
one  will  contend  that  an  infinite  series  will  cause  a  coincidence 
of  M  a  and  D  sooner  than  it  will  the  coincidence  of  C  and  B, 
or,  that  an  infinite  series  of  progressive  steps  will  sooner  ex- 
tinguish a  mean  between  M  a  and  D,  than  between  C  and  B. 

Hence,  C  and  B  must  coincide  at  the  same  time,  and  in  the 
same  point  with  M  a  and  D.  And  as  a  mean  between  A  and  B 
is  also  a  mean  between  C  and  D,  an  infinite  approximation  will 
cause  A  and  B  and  C  and  D  to  coincide  at  the  same  time,  and 
in  the  same  point ;  and  in  respect  to  the  coincidence  of  A  and 
B,  or  of  M  a  and  D,  the  same  is  in  %,. 

But  M  a,  D,  G  and  Z  finally  abide,  and  denote  the  area  and 
linear  measures  of  the  circle ;  and  when  they  coincide  together 
in  ^,  (which  occurs  in  accordance  with  any  law  of  polygonal 
progression,)  the  true  and  mean  places  of  M  a  coincide  in  ^. 

A,  B,  E  and  Y  always  denote  polygons  of  sides,  and  hence 
are  extinguishable  quantities,  or  extinguish  able  means  ;  for  while 
they  are  considered  in  the  nature  of  quantities,  they  are  also  con- 
sidered as  mean  proportionals,  and,  whether  as  quantities  or 
means,  they  are  conceived  to  be  progressing  towards  their  final 
extinguishment,  or  to  a  merger  in  the  abiding  quantities ;  and 
hence,  when  they  cease  to  exist  as  quantities,  they  also  cease  to 
exist  as  means ;  and  by  the  law  of  progression,  A,  B,  E  and  Y 
must  apparently  coincide  in  ^ ;  and  if  either  one  of  them  is  ex- 
tinguished in  ,£,,  they  all  are ;  we  nevertheless  conceive  them  to 
exist  (both  as  quantities  and  means)  in  the  point  of  their  extin- 
guishment ;  and  hence  we  reject  the  conception  of  their  final  ex- 
tinguishment, even  in  the  abiding  quantities  which  finally  denote 
the  circle ;  nor  can  such  conception  be  relieved,  except  by  the 
reductio  ad  absurdum. 


OF    THE    CIRCLE.  79 

• 
SECTION    FIFTH. APPLICATION. 

In  making  a  short  application  of  my  determination  of  the 
measures  of  the  circle  to  that  of  solids,  I  shall  go  no  farther  than 
a  short  comparison  with  the  prime  sphere,  namely,  the  sphere 
whose  diameter  is  unity,  and  its  circumscribed  cylinder  and 
prisms,  amongst  which  I  place  the  cube ;  nor  need  I  go  farther 
than  to  compare  the  prime  sphere  with  its  circumscribing  cylin- 
der and  cube. 

What  is  often  termed  the  volume  or  contents  of  a  solid,  I  shall 
call  by  the  more  appropriate  name  of  bulk. 

The  surfaces  of  regular  solids  are  as  their  bulks. 

The  bulk  of  the  sphere  is  demonstrated  to  be  two-thirds  of  the 
bulk  of  its  circumscribing  cylinder ;  and  the  surface  of  a  sphere 
is  =  to  the  lateral  surface  of  its  circumscribing  cylinder. 

The  surface  of  the  prime  sphere,  or  of  its  circumscribing  cyl- 
inder, or  any  circumscribing  prism,  is  six  times  the  bulk  of  the 
given  solid. 

The  surface  of  a  regular  solid  flows  by  a  duplicate  ratio,  and 
the  bulk  by  a  triplicate  ratio  to  that  of  any  given  linear  measure 
of  such  solid. 

The  circumscribing  cube  of  the  prime  sphere  properly  de- 
notes unity  amongst  the  solids,  as  the  square  does  amongst  the 
polygons. 

The  circumference  of  the  prime  sphere  is  =  the  surface; 
hence,  if  the  diameter  of  a  sphere  be  expressed  in  a  term  of  the 
prime  table,  any  given  power  of  the  circumference  will  be  ex- 
pressed in  terms  of  the  table,  —  as  also  of  the  surface, — 
but  not  of  the  bulk.  Nevertheless,  tables  may  be  readily  made 
upon  the  same  ratio  between  terms  of  the  table,  as  that  of  the 
prime  table,  in  which  the  bulks  of  spheres  may  be  expressed 
when  their  surfaces  are  expressed  in  the  prime  table,  and  in 
which  the  surfaces  of  cylinders  and  their  bulks  may  also  be  ex- 
pressed ;  and  hence  the  whole  may  be  thrown  into  rational  and 
tabular  form. 

It  is  not  my  intention  here  (like  Montucla)  to  enter  into  the 
sports  of  mathematics,  nor  to  make  sport  of  those  things  which 
I  do  not  comprehend  ;  but  I  have  before  suggested  that  number 
8,  which  has  much  to  do  with  the  octagon  and  its  infinity  of 
equal  means,  was  not  destitute  of  beauties ;  —  it  is  the  sum  of  all 
the  powers  of  .8888,  ad  infinitum,  which  is  the  square  root  of  the 
revolving  series  .790123456790,  ad  infinitum,  which  series  does 
not  appear  to  have  a  drop  of  octagonal  blood  in  it.  And  if  either 
series  be  multiplied  or  divided  by  any  natural  number  or  definite 


80  ON   THE    QUADRATURE 

quantity,  the  result  will  be  a  continuous,  uniform,  or  revolving 
series,  by  which  an  infinity  of  equal  means  may  be  produced  as 
exponential  tables. 

These  revolving  series  are  deduced  from  what  may  be  termed 
the  equivalents  of  the  natural  numbers  in  their  order,  of  which  I 
will  give  one  or  two  examples  in  connection  with  a  suggestion  in 
respect  to  the  law  of  falling  bodies.  And  although  I  may  here- 
after give  a  more  extended  numerical  formula  in  respect  to  the 
law  of  falling  bodies,  than  the  one  I  here  give,  and  one  more 
fully  showing  the  economy  and  adaptation  of  numbers  to  our 
every  necessity,  yet,  perhaps  one  of  the  simplest  formulas  that 
may  be  given  for  the  purpose  is  the  following : 

11111111 
1    2    3    4    5    6    7     8    9    10  11  12  13  14  15 

Thus  if  we  take  the  natural  numbers  in  their  order,  to  the 
number  fifteen,  and  divide  them  into  periods  for  extracting  their 
square  root,  making  the  figure  1,  at  the  left,  the  first  period,  the 
square  root  will  come  out  in  the  constant  figure  1,  over  the  last 
figure  in  each  period,  namely,  over  each  odd  number,  making 
eight  figures  in  such  root,  and  the  sum  of  all  the  odd  numbers 
thus  far  will  be  64,  or  the  square  of  the  sum  of  all  the  figures  in 
the  root,  and  so  on,  ad  infinitum,  le't  the  natural  numbers  be  ex- 
tended ever  so  far.  And  in  the  extraction  of  such  root,  the  re- 
mainder will  constantly  come  out  in  terms  of  the  even  numbers 
in  their  order,  as  2,  4,  6,  8,  10,  12,  14,  &c.  Thus  while  each 
figure  of  the  root  denotes  one  moment  of  time,  each  odd  num- 
ber denotes  the  number  of  spaces  passed  over  in  each  successive 
moment,  and  the  successive  even  numbers  denote  the  final  ve- 
locity of  the  body  at  the  end  of  each  successive  moment ;  and 
the  total  spaces  passed  over  at  the  end  of  each  successive  mo- 
ment, will  be  the  square  of  the  number  of  moments  the  body 
has  been  falling. 

Hence,  the  whole  space  passed  over  is  the  square  of  the  whole 
time  of  the  fall ;  and  as  the  force  which  urges  the  body  in  its 
fall  is  constant  and  uniform  with  the  time,  hence,  in  a  falling 
body,  the  whole  space  passed  over  from  the  commencement  of 
the  fall,  is  as  well  the  square  of  the  whole  force,  as  of  the  time, 
which  is  too  important  a  matter  to  be  overlooked  in  our  investi- 
gations of  the  laws  of  force  and  motion,  although  I  apprehend 
it  has  not  always  been  sufficiently  regarded. 

This,  however,  is  one  of  the  numerical  beauties  of  numbers, 
applicable  to  our  wants,  but  we  should  not  seek  to  extend  it  be- 
yond its  legitimate  application. 

But  to  show  that  this  is  a  mere  numerical  dress  of  the  law  of 
falling  bodies,  I  will  present  the  natural  numbers  in  their  order, 


OP    THE    CIRCLE.  81 

or  what  is  equivalent  thereto,  in  another  dress  or  form,  and  which 
may  be  extended  ad  infinitum,  and  which  will  be  found  to  have 
some  application  to  the  measures  of  the  circle,  and  of  the  round 
bodies. 

Thus,  if  we  take  the  natural  numbers  in  their  order  to  any  ex- 
tent, and  multiply  the  same  by  2,  and  then  divide  the  product  by 
2,  we  obtain  a  revolving  series,  ad  infinitum,  which  may  be 
said  to  be  equivalent  to  the  natural  numbers  in  their  order;  which 
series  will  come  out  1,  2,  3,  4,  5,  6,  7,  9,  0,1,  2,  3,  4,  5,  &c.,  in 
which  the  figure  8  will  never  occur. 

If  we  then  divide  such  series  into  periods  for  the  extraction  of 
the  square  root,  making  figure  1,  at  the  left,  one  period,  the  root 
will  continue  to  be  expressed  by  figure  1,  over  the  last  figure  in 
each  period,  ad  infinitum,  and  the  remainder  will  constantly  come 
out  in  terms  of  the  even  numbers,  or  what  is  equivajent  thereto, 
namely,  2,  4,  6,  *9,  1,  3,  5,  8,  0,  2,  4,  6,  9,  1,  3,  &c.,  which  equiv- 
alent of  the  even  numbers  is  produced  in  the  same  manner  as 
that  of  all  the  natural  numbers.  So  also  an  equivalent  to  the 
odd  numbers  is  produced  in  the  same  manner,  and  will  revolve 
thus :  1,  3,  5,  8,  0,  2,  4,  6,  9,  1,  3,  5,  8,  0,  2,  &c.  Which  equiva- 
lents of  the  natural  numbers  in  their  order,  when  applied  to  the 
measures  of  the  circle  and  of  the  round  bodies,  are  not  only  use- 
ful, but  their  application  is  extremely  beautiful ;  but  they  do  not 
so  well  apply  as  formulas  to  express  the  law  of  falling  bodies 
near  the  earth. 

But  to  return  to  a  farther  application  of  the  circle.  When  the 
diameter  of  the  inscribed  sphere  of  the  cylinder  is  =  d  of  the 
circle  whose  m  is  1,  the  bulk  of  such  cylinder  is  the  square  of  the 
diameter  of  the  inscribed  sphere.  T-ie  circumference  is  four,  and 
the  surface  will  be  six  times  the  diameter  of  its  inscribed  sphere. 
But  the  proper  diameter  of  the  cylinder  or  cube,  is  the  diameter 
of  its  inscribed  sphere.  If  the  diameter  of  the  cylinder  be  the 
sixth  root  of  six,  its  surface  is  six,  and  its  bulk  is  =  its  diameter. 

But  this  is  the  general  law :  wher  the  diameter  of  the  sphere, 
cylinder  or  cube  is  the  reciprocal  of  '  e  bulk  of  a  like  prime  solid, 
the  bulk  is  the  square  of  the  diameter.  Hence,  when  the  diam- 
eter of  the  sphere  is  1.889881,  or  th?rd  root  of  6.75,  the  bulk  is 
the  square  of  the  diameter,  and  the  \rface  is  six  times  the  diam- 
eter, and  the  circumference  is  6.  So  when  the  diameter  of 
such  solid  is  the  third  root  of  the  reciprocal  of  the  bulk  of  a  like 
prime  solid,  the  bulk  is  1.  The  reciprocal  of  the  bulk  of  the 
prime  sphere  is  three  times  the  square  of  M  of  the  prime  circle, 
or  is  three  times  three  halves  of  M  01  the  prime  circle,  and  the 
square  of  such  reciprocal  of  the  bulk  of  the  prime  sphere,  is  the 
third  root  of  the  square  of  M  3. 
11 


82  ON    THE    QUADRATURE 

The  square  root  of  the  reciprocal  of  the  bulk  of  a  prime  solid 
is  the  point  of  equality  between  the  bulk  and  diameter,  and  the 
third  root  of  the  square  of  such  point  of  equality  is  the  diameter 
of  a  like  solid  whose  bulk  is  1 ;  and  such  point  of  equality  in 
the  prism  of  three  lateral  sides  is  ==  B  6. 

.888,  ad  infinitum,  is  six  times  the  third  power  of  the  bulk  of 
the  prime  sphere,  and  is  =  the  area  of  the  circle  whose  m  is  "jp 
above  .666,  ad  infinitum^  and  is  three  halves  of  the  bulk  of  the 
prism  of  three  lateral  sides,  whose  diameter  is  =  d  3,  when  M  3 
is  1,  &c. 

The  reciprocal  of  the  bulk  of  any  prime  solid  is  =  the  bulk  of 
the  circumscribing  cube  of  a  like  solid  whose  bulk  is  1.  But  I 
will  not  pursue  the  subject  farther ;  the  object  being  simply  to 
show  the  reciprocities  and  coincidences,  and  the  mathematical 
beauties  in  which  solids  seem  inclined  to  clothe  themselves. 

The  father  of  true  physical  philosophy  has  .shown  that  the 
squares  of  the  periods  of  the  planets  are  as  the  cubes  of  their 
mean  distances  from  the  sun  ;  and  wishing  to  be  justified  or 
corroborated  in  my  determination  of  the  quadrature,  by  that  great 
physical  law  in  regard  to  force  and  motion,  I  will  here  make  a 
reference  of  the  consequences  of  my  determination,  to  the  two 
great  Keplerean  laws  ;  and,  in  such  reference,  shall  (for  the  pres- 
ent purposes)  consider  orbits  of  different  dimensions  as  being 
circles  of  different  dimensions  ;  premising  that  while  the  circle 
whose  diameter  is  unity  is  the  prime  circle,  the  orbit  whose  ra- 
dius is  unity  is  properly  the  prime  orbit ;  and  this  for  reasons 
that  will  readily  be  recognized.  And  I  will  here  remark,  that  in 
general  we  have  as  little  to  do  with  the  area  of  an  orbit  in  the 
investigation  of  the  laws  of  force  and  motion,  as  we  have  with 
the  areas  of  polygons  in  the  investigation  of  the  measures  of  the 
circle. 

Unity  is  the  standard  measure  for  anything  susceptible  of  nu- 
merical admeasurement,  or  of  ratio  and  proportion  ;  hence,  when 
the  radius  of  an  orbit,  or  the  mean  distance  of  the  planet  is  1,  the 
period  is  1 ;  else  the  square  of  the  period  could  not  be  as  the 
cube  of  the  distance  ;  so  when  radius  is  1,  the  force  of  gravity 
must  be  called  1, — for  such  is  the  only  condition  upon  which 
we  can  proportion  the  rate  of  gravity  to  different  distances  from 
the  same  attracting  power.  So  also,  at  the  distance  1,  the  mean 
rate  of  motion  is  1,  as  also  the  rate  of  convergency  of  the  orbit, 
and  especially  the  time  of  the  period. 

I  will  here  assume  the  fact,  that  if  the  diameter  of  the  inscribed 
circle  of  any  given  polygon  be  =  to  the  square  of  the  circum- 
ference of  a  like  major  polygon,  the  circumference  of  such  poly- 
gon will  be  the  third  power  of  the  circumference  of  such  major 


OP    THE    CIRCLE.  83 

polygon  ;  and  the  same  will  hold  good  in  respect  to  the  circle,  or 
orbit. 

Hence,  when  the  diameter  of  an  orbit  is  =  the  square  of  the 
circumference  of  the  circle  whose  diameter  is  1,  the  circumfer- 
ence of  such  orbit  will  be  32,  or  one  fourth  of  the  circumference 
will  be  8. 

Or  thus  :  when  the  radius  of  an  orbit  is  four  times  the  square 
of  the  circumference  of  the  prime  circle,  the  square  of  the  cir- 
cumference of  such  orbit  will  be  the  third  power  of  radius  (the 
mean  distance)  of  such  orbit ;  and  such  necessarily  occurs  when 
the  circumference  of  the  orbit  is  256,  in  which  case,  the  period 
will  be  =  the  circumference. 

Or  the  version  may  be  thus :  —  if  the  circumference  of  ah 
orbit  be  the  product  of  the  third  power  of  the  circumference  of 
the  prime  circle,  by  8,  the  period  will  be  =  the  circumference-— 
for  when  the  diameter  of  an  orbit  is  8,  the  period  is  =  the  diam- 
eter, by  any  system  of  the  circle. 

So  when  the  diameter  of  an  orbit  is  8,  the  circumference  is 
twice  the  square  of  the  circumference  of  the  square  (or  p  4)  of 
equal  area  with  that  of  the  prime  circle. 

The  third  power  of  the  circumference  of  an  orbit  is  32  times 
the  third  power  of  the  diameter,  and  256  times  the  third  power 
of  the  radius ;  so  when  the  circumference  is  256,  the  square  of 
the  circumference  is  the  cube  of  radius. 

So  when  the  circumference  is  64,  the  period  will  be  32,  and 
when  the  circumference  is  16,  the  period  will  be  4,  and  so  on. 
So  when  the  circumference  is  4,  the  period  is  one  eighth  of  the 
circumference ;  viz.,  the  period  will  be  .5. 

Thus,  when  the  radius  of  an  orbit  is  the  third  root  of  any 
given  point,  the  period  of  the  planet  is  the  second,  or  square  root 
of  the  same  point.  And  hence  we  may  conclude  that  this  great 
law  is  founded  in  unity  of  purpose. 

Suppose  then  a  planet  to  revolve  in  a  centric  orbit,  namely 
with  the  sun  in  the  centre,  and  of  course  having  a  uniform  mo- 
tion :  Then  if  the  force  of  gravity  be  inversely  as  the  distance, 
(as  I  allege  it  to  be,)  the  rate  of  motion  will  be  the  square  root 
of  the  force  applied ;  but  if  the  force  of  gravity  varies  inversely 
as  the  square  of  the  distance,  as  taught  by  Sir  Isaac  Newton, 
then  the  rate  of  motion  will  be  the  fourth  root  of  the  force  appli- 
ed ;  as  may  readily  be  seen  by  almost  any  one  who  will  bestow 
a  little  consideration  on  the  subject.  The  circumferences  of 
circles  are  as  their  diameters,  or  as  their  radii ; — hence,  if  Kepler, 
could  have  supposed  that  his  great  law,  as  by  him  declared, 
could  have  been  mistaken  or  perverted,  he  might  have  declared 
anew,  that  the  third  power  of  one  eighth  of  the  circumference 


84  ON    THE    QUADRATURE 

is  half  the  square  of  the  period ;  or  that  the  square  of  twice 
the  period  is  the  third  power  of  one  fourth  of  the  circumference, 
&c.j  thus  comparing  time  with  circumference,  in  lieu  of  di- 
ameter; and  the  law  thus  declared  would  have  been  equally 
true  and  equally  well  received.  For  then,  as  now,  when 
the  circumference  is  4,  the  period  will  be  .0,  or  equal  to  one- 
eighth  of  the  circumference ;  and  still,  when  the  diameter  is  8, 
the  period  will  be  8,  and  when  the  period  is  4,  it  will  be  =  one- 
fourth  of  the  circumference,  or  the  square  root  of  the  circumfer- 
ence, &c. 

When  the  period  is  .5,  or  the  square  root  of  .25,  the  radius  is 
necessarily  the  cube  root  of  .25,  or  the  square  of  3>  and  the  di- 
ameter must  be  the  reciprocal  of  ^  by  any  system ;  else  the 
Keplerean  system  must  fail. 

This  statement,  however,  will  not  quadrate  with  the  popular 
measures  of  the  circle ;  and  it  will  be  seen  and  acknowledged, 
that  in  applying  the  popular  measures  of  the  circle  to  the  laws 
of  planetary  force  and  motion,  it  requires  a  great  deal  of  calculi 
to  chink  up  with.  According  to  this,  the  third  power  of  one- 
fourth  of  the  circumference  of  an  orbit  when  the  diameter  is  1, 
is  equal  to  the  period  of  a  planet  in  an  orbit  whose  circumference 
is  4;  —  but  not  so  says  the  popular  method. 

By  any  system,  the  rate  of  motion,  and  the  period  of  a  planet 
have  a  determinate  ratio  to  the  diameter  of  the  orbit ;  but  accord- 
ing to  the  popular  method,  they  have  no  determinate  ratio  to 
the  circumferential  measure  of  the  orbit;  that  is,  according  to 
the  popular  notions,  neither  the  motion  nor  time  of  the  period, 
have  a  determinate  ratio  to  that  of  the  line  or  path  in  which  the 
planet  actually  moves  or  revolves. 

By  any  system,  when  the  circumference  is  4,  or  when  the  area 
is  =  the  diameter  of  the  orbit,  the  period  is  the  third  power  of 
the  diameter  of  an  orbit  whose  area  is  .5, —  hence,  the  period  of 
any  planet  of  a  system  is  =  the  third  power  of  the  diameter  of 
another  orbit  of  the  system  whose  area  is  =. one-eighth  of  the 
circumference  of  the  given  orbit.  Thus,  if  the  circumference  of 
an  orbit  be  256,  the  period  is  =  the  third  power  of  the  diameter 
of  an  orbit  wh  se  area  is  32. 

This  application  may  also  be  made  to  orbits  in  connection 
with  polygons,  namely,  the  reciprocal  of  M  of  any  given  poly- 
gon is  the  diameter  of  an  orbit  in  which  the  period  is  the  third 
power  or  cube  of  the  diameter  of  the  circumscribed  circle  of  a 
like  polygon  whose  area  is  .5. 

If  the  limits  assigned  to  this  work  permitted,  it  would  by  no 
means  be  an  unpleasant  task  to  contemplate  the  relationship  of 
the  family  of  curves ;  and  wherein  the  parabola  (that  curve  of 
all  others  except  that  of  the  circle,  the  most  useful  and  impor- 


OF    THE    CIRCLE.  85 

tant  to  be  understood,)  conspires  with  the  circle  in  the  admeas- 
urement of  solids,  &c.,  and  serves  to  corroborate  my  determina- 
tion of  the  quadrature. 

It  is  entirely  beyond  my  sagacity  to  determine  how  the  ellipse 
(or  elongated  circle)  has  been  elaborated  into  notice  and  conse- 
quence far  beyond  its  merits,  otherwise  than  by  way  of  mathe- 
matical or  geometrical  discipline  to  the  mind ;  and  I  am  equally 
uninformed  why  the  science  termed  the  science  of  conic  sec- 
tions, (from  wh^ch  the  modest  circle  is  generally  excluded,)  has 
been  regarded  paramount  to  any  other  department ;  —  of  which 
science  it  is  said,  by  those  in  high  places,  that  "  it  forms  one  of 
the  most  important  parts  of  mathematics,  being  distinguished 
for  elegance,  and  demonstrating,  with  surprising  simplicity  and 
beauty,  and  in  the  most  harmonious  connection,  the  different 
laws,  according  to  which  the  Creator  has  made  worlds  to  re- 
volve," &c.  — "  that  few  branches  of  mathematics  delight  a 
youthful  mind  so  much  as  conic  sections ;  that  the  emotions 
which  the  pupil  manifests  when  they  unfold  to  him  the  great 
laws  of  the  universe,  might  be  called  natural  piety ;  and  that  in 
teaching  this  science  to  young  people,  the  descriptive  method 
(by  diagrams)  ought  always  to  be  connected  with  the  analytic 
method,"  &c. 

I  am  aware  that  the  ellipse,  the  parabola,  and  the  hyperbola, 
may  be  presented  in  a  train  of  diagrams  which  in  their  appear- 
ance are  very  imposing ;  and  that  they  may  serve  in  connection 
with  the  importance  which  is  generally  supposed  to  be  attached 
to  their  explication,  to  induce  a  species  of  faith  or  belief,  which 
some  would  call  natural  piety.  But  I  have  yet  to  learn  that 
this  science  discloses  the  laws  according"  to  which  the  Creator 
has  made  worlds  to  revolve  ;  or  that,  —  as  has  also  been  admir- 
ingly expressed,  —  in  comparison  with  conic  sections,  in  view 
of  the  grandeur  and  beauty  which  they  unfold  to  the  mind  in 
the  mathematical  world,  all  that  has  preceded  this  part  is  but  the 
alphabet  of  the  science  of  mathematics,  &c. 

It  is  true  that  the  strong  grounds  thus  far,  of  the  Newtonian 
philosophy,  is  in  the  belief  that  he  disclosed,  by  their  aid,  the 
law  according  to 'which  the  Creator  has  made  worlds  to  revolve; 
and  so  tenacious  have  been  the  sustainers  of  Newton's  fame, 
that  the  cry  of  robbery  was  raised  when  Bernoulli  made  the 
allegation  without  giving  Newton  the  credit,  that  "we  now  know 
that  the  heavenly  bodies  perform  their  revolutions  by  conic  sec- 
tions ;"  thereby  inducing  the  world  to  attach  vast  importance  to 
what  may,  at  some  future  day,  be  shown  to  be  wholly  delusive. 

It  is  somewhat  curious  to  contemplate  the  means  and  the  in- 
genuity which  have  been  practised,  to  induce  the  natural  piety 


86  ON    THE    QUADRATURE 

conceived  to  flow  from  a  contemplation  of  conic  sections  ;  and 
it  is  certain  that  those  articles  of  faith  have  been  fortified  by 
great  care  and  caution,  and  by  a  continual  inculcation  ;  so  much 
so,  that  not  a  word  must  be  suggested,  in  our  works  on  astron- 
omy, that  it  would  be  possible  for  a  planet  to  revolve  in  a  circle, 
whether  centric  or  eccentric ;  nor  that  the  ellipse  can  be  cut  from 
any  other  solid  than  the  cone.  Hence,  the  natural  minds  of 
thousands  have  been  so  continually  plied  with  the  elliptical  or- 
bit, and  with  the  magic  qualities  of  the  cone  and  of  conic  sec- 
tions, that  at  length  they  have  yielded,  or,  at  least,  rendered  pas- 
sive obedience.  And  it  would  not  be  wonderful  if  something 
strange  should  come  over  the  mind,  when  it  is  brought  to  assent 
to  the  Newtonian  hypothesis,  that  the  Creator  has  caused  worlds 
to  revolve  in  ellipses,  however  eccentric. 

Sir  Isaac  Newton  settled  many  knotty  questons  by  his  favor- 
ite expression,  of  Neque  novit  natura  limitem,  which,  perhaps, 
may  have  various  renderings,  for  it  will  as  well  apply  to  error  as 
to  truth  ;  for  when  we  have  set  out  on  the  road  to  error,  we  can 
find  no  bounds  or  limits  to  our  career ;  and  such  is,  doubtless, 
the  case  in  respect  to  the  hypothesis  of  elliptical  orbits,  —  as  I 
shall  attempt  to  prove,  —  an  error  which  commenced  with  Kep- 
ler, while  making  calculations  upon  Tycho's  observations ;  an 
error  which,  although  not  depending  upon  any  physical  necessity 
or  law,  was  seized  upon  by  Newton ;  and  he  has  given  it  such 
importance,  as  nearly  to  obliterate  the  two  great  laws  of  Kepler, 
and  especially  that  which  declares  that  the  squares  of  the  periods 
are  as  the  cubes  of  the  mean  distances,  —  the  modern  version 
of  which  is,  that  "  the  squares  of  the  periods  of  the  planets  are 
as  the  cubes  of  their  semi  major  axes." 

If  such  then  be  the  law,  —  namely,  that  a  planet  may  revolve 
in  an  orbit,  however  eccentric,  even  to  extreme  eccentricity, 
and  that  the  square  of  the  period  is  the  cube  of  the  semi  major 
axis,  —  it  is  manifest,  that  during  the  period  of  such  planet,  a 
vastly  greater  amount  of  the  force  of  attraction  will  be  applied 
to  it,  than  to  a  planet  revolving  in  a  circle,  (whether  centric  or 
eccentric,)  whose  diameter  is  =  to  the  major  axis  of  the  ellipse ; 
while  the  planet  revolving  in  the  ellipse  would  pass  over  less 
than  two-thirds  of  the  space  of  that  revolving  in  the  circle,  in 
the  same  time  or  period,  and  a  far  less  proportion  of  area. 
Hence,  the  popular  notion  makes  the  time  of  the  period  depend 
wholly  upon  the  major  axis  of  the  orbit,  without  regard  to 
amount  of  space  passed  over,  or  the  amount  of  force  expended 
during  the  period.  And  such  have  been  the  doctrines  and  teach- 
ings which  have  induced  what  is  termed  natural  piety,  in  those 
who  have  been  induced  to  conceive  that  such  things  could  be. 


OF    THE    CIRCLE.  87 

If  it  shall  be  found,  then,  that  a  planet  cannot  revolve  in  an 
orbit  of  extreme  ellipticity,  if  we  retrace  the  road  which  led  to 
such  error,  we  shall  find  that  a  planet  cannot  revolve  in  an  el- 
liptical orbit,  however  small  the  ellipticity. 

I  will,  however,  in  conclusion  of  this  reference,  suggest  a  dic- 
tum or  two,  based  upon  the  operation  of  the  Keplerean  law,  — 
namely,  that  the  square  of  the  period  is  the  cube  of  the  distance, 
and  that  the  radius  vector  passes  over  equal  areas  in  equal  times  ; 
vconsequently,  that  the  motion  of  the  planet,  at  any  given  point  of 
the  orbit,  varies  inversely  as  the  distance  varies, — which  laws,  in 
fact,  are  one  and  the  same,  operating  in  unity  of  purpose,  although 
the  same  has  generally  been  resolved  into  two  separate  or  dis- 
tinct laws.  This  law,  then,  taken  in  connection,  presupposes  its 
operation  in  an  eccentric  orbit,  or  one  in  which  the  central  force 
is  not  situated  in  the  centre  of  the  orbit. 

This  subject,  however,  has  been  dark,  mysterious  and  inexpli- 
cable, in  consequence  of  the  Newtonian  error,  in  assigning  the 
law ;  and  notwithstanding  it  had  become  much  the  fashion  im- 
plicitly to  avow  that  to  be  true  which  it  was  said  Sir  Isaac  New- 
ton had  found,  nevertheless,  for  many  years  after  the  promulga- 
tion of  the  Newtonian  law  of  gravity,  the  truth  had  not  become 
so  far  entangled  in  the  meshes  of  error,  as  to  place  it  wholly 
beyond  the  power  of  redemption,  until  Clairaut  effectually  accom- 
plished it,  in  his  supposed  accounting  for  the  motion  of  the 
moon's  apogee,  upon  the  principle  of  the  Newtonian  law  of 
gravity ;  for,  previous  to  that  time,  it  had  not  been  deemed  high 
treason  for  men  in  high  places  to  speak  disparagingly  of  the  law. 

But  still  the  square  of  the  period  is  the  cube  of  the  mean 
distance,  and  still  the  motion  varies  inversely  as  the  distance 
varies,  and  consequently  the  line  of  the  radius  vector  passes  over 
=  ' areas  in  equal  times  ;  and,  consequently,  the  simplicity  in  the 
laws  of  force  and  motion,  by  which  the  heavenly  bodies  revolve, 
should  not  have  been  rendered  thus  complicated  and  inexplica- 
ble. For  the  above  phenomena  once  being  established,  directly 
and  conclusively  determine  the  law  of  gravity,  by  which  the  phe- 
nomena are  caused  or  produced  ;  that  is,  the  law  is  directly  deduc- 
ible  from  its  effects  ;  and  why  it  should  not  have  been  done,  is 
somewhat  unaccountable. 

I  allege,  then,  that  the  intensity  of  gravity  is  inversely  as  the 
distance,  and,  consequently,  that  the  mean  motion  of  a  planet  is 
the  square  root  of  the  mean  force ;  (not  that  the  mean  motion  is 
the  fourth  root  of  the  mean  force,  as  Sir  Isaac  Newton  would 
have  it.) 

Then,  as  the  motion  of  a  planet  in  an  eccentric  orbit  varies 
inversely  as  the  distance  varies,  hence  the  motion,  at  any  given 


88  ON  THE  QUADRATURE 

point  of  the  orbit,  will  be  the  square  of  what  the  motion  of  a 
planet  would  be,  in  a  corresponding  point  of  a  like  orbit,  whose 
mean  distance  is  the  square  root  of  the  mean  distance  of  the 
given  orbit;  and  this  for  the  reason,  that  the  mean  motion  of  a 
planet  is  inversely  as  the  square  root  of  the  mean  distance ; 
else  the  squares  of  the  periods  of  a  system  of  planets  cannot 
be  as  the  cubes  of  their  mean  distances  from  the  gravitating 
power. 

Hence,  if  the  mean  distance  be  4,  and  the  perihelion  distance- 
1,  the  actual  motion,  at  any  given  point  of  the  orbit,  is  the  square 
of  what  the  actual  motion  of  a  planet,  revolving  in  a  centric  or- 
bit, would  be,  at  just  half  the  distance  of  such  given  point. 
Thus,  if  the  mean  distance  be  4,  the  mean  motion  will  be  .5 ;  or 
the  square  of  the  motion  of  a  planet,  revolving  around  the  same 
central  force  in  a  centric  orbit,  at  the  distance  2,  or  at  half  the 
mean  distance  of  the  given  orbit.  And  if  the  perihelion  distance 
of  such  orbit  be  1,  the  motion  at  the  perihelion  would  be  2,  or 
the  square  of  the  motion  of  a  planet,  revolving  in  a  centric  orbit 
about  the  same  central  force,  at  the  distance  .5,  or  half  the  dis- 
tance of  the  given  planet  at  perihelion,  and  the  like  would  be  the 
case  in  any  point  of  the  orbit. 

And  such  will  be  found  to  be  some- of  the  necessary  and  inev- 
itable results  of  the  law  of  gravity,  and  they  are  in  perfect  ac- 
cordance with  the  great  phenomena  observed  by  Kepler,  namely, 
that  the  squares  of  the  periods  are  as  the  cubes  of  the  distances ; 
that  the  motion  varies  inversely  as  the  distance  varies  ;  and,  con- 
sequently, that  the  radius  vector  passes  over  equal  areas  in  equal 
times,  and  that  none  of  these  phenomena  can  occur,  except  the 
force  of  gravity  be  inversely  as  the  distance. 

SECTION    SIXTH. 

Infallibility  is  certainly  a  rare  quality  to  acquire  during  the 
whole  period  of  one's  life,  and  but  few  may  expect  to  be  charged 
or  credited  with  it ;  and  I  certainly  have  no  desire  that  the  world 
should  enter  upon  their  books  any  such  charges  or  credits  to  me ; 
and  should  I  be  charged  with  an  attempt  to  mar  the  record  of 
the  world  in  respect  to  the  infallibility  of  Sir  Isaac  Newton  and 
some  others,  for  whom  the  votaries  of  science  have  long  claimed 
the  entire  admiration  of  the  world,  I  have  only  to  remark  that 
such  admiration  has  savored  quite  too  much  of  idolatry  to  sub- 
serve the  best  interests  of  science ;  and  I  have  never  yet  believed 
that  the  only  proper  way  to  extend  or  perfect  science  in  the  eyes 
of  a  wondering  or  admiring  world,  is,  by  inducing  a  belief  in  the 
infallibility  of  some  one. of  its  votaries  who  may  have  essayed,  to 


OP   THE    CIRCLE.  O» 

lay  its  broad  foundations.  And  having  honestly  conceived  that 
certain  principles  promulgated  by  Sir  Isaac  Newton  as  funda- 
mental, (and  so  received  and  taught  b^  the  world,)  are  wholly 
erroneous,  —  and  that  long  before  this  the  errors  would  have 
been  detected  and  repudiated,  but  for  the  admiration  of  the  sup- 
posed infallibility  of  Sir  Isaac  Newton,  so  diligently  and  success- 
fully inculcated,  —  I  have  not  hesitated  to  attempt  a  reformation, 
by  pointing  out  such  defects  and  want  of  comeliness  in  the  ob- 
ject of  adoration,  as  I  supposed  could  not  well  be  overlooked, — 
even  though  the  charge  of  jealousy  of  the  fame  of  Sir  Isaac 
Newton  should  be  as  industriously  sounded  abroad  against  my- 
self as  it  has  been  against  Liebnitz,  Hook,  Bernoulli,  and  others  of 
Newton's  cotemporaries,  who  were  to  be  properly  disposed  of,  that 
Newton  might  be  regarded  by  the  world  as  "  the  creator  of  nat- 
ural philosophy."  For  such  has  been  the  constant  practice  of 
the  world ;  nor  have  our  ears  been  in  anywise  relieved  from  the 
trump  of  fame,  notwithstanding  the  modern  disclosures  in  re- 
spect to  his  treatment  of  Flamstead,  and  the  indignation  conse- 
quent thereon,  to  allay  which  has  even  required  the  able  pen  of 
Mr.  Whewell. 

Thus,  in  most  English  treatises  upon  science,  we  are  con- 
stantly reminded  of  the  quarrel  between  Liebnitz  and  Newton, 
in  respect  to  priority  of  discovery  of  fluxions,  or  the  calculus, 
considered  so  important  in  the  involution  and  evolution  of  curves, 
&c.,  which,  after  having  been  introduced  by  the  celebrated  Mr. 
Huygens  in  the  legitimate  application  of  the  cycloid  to  pendu- 
lums of  clocks,  has  certainly  been  carried  to  great  lengths  in 
speculative  philosophy,  even  to  those  doubts  and  difficulties 
which  are  acknowledged  to  be  almost  inexplicable,  and  which, 
perhaps,  of  all  the  branches  of  science,  best  serve  to  give  the  his- 
tory of  the  war  between  Newton  and  Liebnitz,  which  is  often 
set  forth  in  such  descriptions  as  might  enable  a  Fuseli  to  depict 
those  giants,  each  having  hold  of  a  horn  of  the  dilemma,  and 
contending  manfully  for  the  milk.  But  I  protest,  that  it  is  both 
the  privilege  and  the  duty  of  every  free  born  son  of  earth,  to  be 
jealous  of  the  fame  of  any  one,  if  such  fame  shall  tend  to  check 
or  stop  the  upward  progress  of  man,  either  moral  or  scientific,  to 
which  he  seems  destined  by  a  proper  development  of  his  capa- 
cities. 

It  is  said,  however,  that  "  the  evolution  and  involution  of 
curved  lines  have  great  claims  on  our  attention  ; "  and  that  "  this 
doctrine  is  peculiarly  valuable  to  the  speculator  in  the  higher 
mechanics."  And  in  fact,  we  find  that  the  thread  of  involution 
and  evolution  has  been  industriously  interwoven  into  the  me- 
chanics of  the  heavens,  by  ingenious  processes,  which  are  quite 
12 


90  ON    THE    QUADRATURE 

surpassing  common  comprehension,  and  therefore  destined  to  re- 
main, except  to  the  initiated,  as  one  of  the  mysteries  of  science. 

Nevertheless,  one  great  object  of  this  branch  of  science  is, 
to  ascertain  the  flexure  or  variation  of  curvature  in  any  given 
point  of  any  curve  other  than  that  of  the  circle,  (as  that  of  the 
parabola,  for  instance,)  when  compared  with  the  uniform  curva- 
ture of  the  circle,  whose  magnitude  is  so  adjusted  to  the  occasion 
as  to  give  it  the  name  of  the  equi- curve  circle ;  in  which  case  the 
curvature  of  such  circle  is  conceived,  or  supposed,  to  be  equal  to 
the  curvature  of  the  parabola  in  the  given  point.  And  it  is  said, 
"  had  Appollonius  but  noticed  a  single  fact,  he  would  have  dis- 
covered the  whole  theory  of  evolution,  and  its  importance  in 
speculative  geometry. 

Thus,  in  the  case  of  involution  and  evolution,  as  in  the  case 
of  conic  sections,  the  circle  is  considered  to  be  the  standard 
measure,  to  which  other  curves  must  be  referred  in  our  investi- 
gations ;  and  hence,  if  there  be  error  in  assigning  the  ratio  of  the 
diameter  of  the  circle  to  the  circumference,  such  error  is  neces- 
sarily visited  upon  every  other  curve  of  which  the  circle  is  neces- 
sarily the  standard. 

The  theory,  then,  of  involution  and  evolution  of  curve  lines, 
as  delivered  by  Mr.  Huygins,  (and  which  is  said  to  have  been 
highly  prized  by  Newton,  who  applied  it  to  trochoids  and  epicy- 
cles of  all  kinds,  but  which  is  said  to  have  been  finally  eclipsed' 
by  Newton's  fluxionary  geometry,)  may  properly  be  said  to  have 
been  the  progenitor  of  fluxions,  and  is  said  to  be  peculiarly 
adapted  to  fluxionary  equations,  by  which  the  fancy  may  con- 
ceive infinities  and  infinitesimals  to  such  an  extent,  that  the  great- 
est of  a  never-ending  series  of  one  kind  of  magnitudes,  shall  be 
infinitely  less  than  the  smallest  of  another  successive  series.  This 
extension  of  nothingness  has,  however,  given  occasion  for  such 
eminent  metaphysicians  as  Newton,  Liebnitz,  Bernoulli,  &c.,  to 
accuse  each  other  of  giving  explanations  of  orders  of  curvature 
that  can  have  no  existence. 

Nevertheless,  whoever  shall  carefully  examine  the  subject  will 
perhaps  perceive  but  little  actual  difference  in  the  extension  of 
their  respective  imaginations  :  All  agree  that  there  is  a  differ- 
ence between  the  simple  contact  of  two  curves,  and  the  point  in 
which  the  circle  shall  osculate  a  curve,  or  the  point  of  osculation  ; 
and  hence,  that  there  can  be  a  closer  union  between  the  curve  of 
the  circle  and  another  curve,  than  that  produced  by  simple  contact. 

Liebnitz  speaks  of  different  degrees  or  orders  of  osculation, 
each  of  which  is  infinitely  closer  than  the  other;  and  it  is  con- 
tended, in  honor  of  Newton,  that  he  had  done  this  before,  &c. ; 
and  that  Liebnitz's  allegation,  that  an  osculation  produced  in  the 
evolution  of  a  curve  is  equivalent  to  four  intersections,  is  alto- 


OF    THE    CIRCLE.  91 

gether  erroneous,  for  it  is  contended  "  that  the  circle  indicated  by 
the  coalescence  of  three  intersections  is  properly  named  the  equi- 
curve  circle ;  that  since  we  measure  all  curvatures  by  that  of  the 
circle,  such  circle  is  properly  the  circle  of  curvature,  and  its  radius 
is  the  radius  of  curvature ;  for  the  flexure  of  the  circle  being  the 
same  in  every  part,  it  becomes  the  proper  index  for  all  curves," 
&c.  Hence  it  is  said  that  "  where  two  intersections  coalesce, 
there  is  a  common  tangent ;  and  when  three  intersections  coalesce, 
there  is  an  equal  curvature,  —  for  then  no  other  circle  can  pass 
between  this  circle  and  the  curve." 

But  it  is  again  and  again  alleged  "  that  there  may  be  indefi- 
nite degrees  of  this  coalescence  of  closeness  of  contact  between 
a  curve  and  the  circle  ;  and  that  this  gradation  of  more  and  more 
intimate  contact,  or  approximation  to  coalescence,  may  be  contin- 
ued without  end  :  Neque  novit  natura  limatem."  And  such  has 
been  the  extent  to  which  this  perplexed  metaphysical  reasoning 
has  been  carried,  that  evolutrixes  have  even  been  conceived  or 
imagined,  which  coincide  with  straight  lines,  and  others  of  in- 
finitely greater  rectitude  which  still  are  curves. 

Now,  however  well  such  a  course  of  reasoning  may  serve  as 
an  exercise  for  the  metaphysician  or  speculative  geometer,  it  may 
nevertheless  appear  to  most  people  of  plain  common  sense,  illy 
adapted  to  any  beneficial  purpose  or  practical  use,  and  may  re- 
mind one  of  the  eulogy  pronounced  upon  Newton's  defence  of 
his  theory  of  optics,  or  analysis  of  light,  against  the  attacks  of 
Hook,  Huygins  and  others,  in  which  it  is  said  his  minuteness  of 
detail  was  so  inconceivably  beyond  the  ken  of  mortals,  as  that 
very  few  were  able  to  comprehend  his  reasoning. 

And  it  may  be  doubted  whether  those  who  contend  for  a 
never-ending  approximation  to  a  closer  and  closer  contact  or  co- 
alescence, than  that  of  a  simple  contact,  have  not  left  the  sub- 
stance to  grasp  at  the  shadow,  and  thereby  extended  their 
conceptions  beyond  the  reality  of  things ;  for  if  a  point  be  but 
position  without  length  or  breadth,  and  if  a  line  be  but  exten- 
sion without  breadth,  it  may  be  hard  to  conceive  how  two  points 
or  lines  can  approximate  nearer  to  each  other  than  simple  contact. 

But  such  fantastic  reasoning  seems  rather  calculated  to  oper- 
ate as  conviction  to  the  world,  than  as  legitimate  proof  in  the 
premises ;  for  in  view  of  the  doubts  and  perplexities  of  those 
who  have  essayed  to  give  the  evidence,-  (and  their  mutual  jeal- 
ousies forbade  their  settling  the  matter  by  convention,)  it  is 
doubtful  whether  a  great  portion  of  mankind  will  undertake  a 
reinvestigation  of  the  subject;  whether  they  will  not  rather  pre- 
fer to  admit  such  reasoning  to  be  beyond  their  comprehension. 
And  hence  the  world  at  large  (with  few  exceptions)  assent  that 
such  may  be  an  exposition  of  those  laws,  "  according  to  which 


92  ON    THE    QUADRATURE 

the  Creator  has  caused  worlds  to  revolve,"  in  accordance  with 
the  Newtonian  theory  and  law  of  gravity,  namely,  by  the  sole 
and  constant  operation  of  disturbing  forces ;  and  hence  it  is  said, 
and  generally  believed,  "  that  all  these  curves  and  angles  of 
contact,  may  be,  and  certainly  are,  every  day  described  by  bodies 
moving  in  free  space,  and  acted  on  by  accelerating  forces  di- 
rected to  different  bodies." 

Those  conclusions,  arising  from  attempts  to  fathom  infinity 
by  progressive  steps,  and  in  which  we  are  as  liable  to  pass  the 
goal  as  to  fall  short  of  it,  have  doubtless  led  to  as  much  of  error 
as  they  have  of  curious  and  unprofitable  speculation.  They 
have  entangled  the  theory  and  law  of  gravity  in  those  meshes 
from  which  the  world  has  been  unable  to  extricate  them ;  and 
although  those  mysteries  seemed  neither  designed  nor  calculated 
to  develope  or  disclose  the  true  theory  or  law  of  gravity,  they 
have  nevertheless  been  supposed  (some  how  or  other)  to  estab- 
lish them  according  to  Newton.  Perhaps  the  careless  or  unskill- 
ed observer  would  never  discover  any  reason  why  a  piece  of 
complicated  machinery  might  not  possess  the  power  of  pro- 
ducing perpetual  motion,  if  it  would  but  go ;  but  Newton  has 
wholly  the  advantage  in  the  parable,  for,  thank  heaven,  the  sys- 
tem of  worlds  continues  to  move,  however  rickety  it  may  have 
become  in  consequence  of  having  been  transferred  from  the  laws 
of  Nature  to  those  of  Newton. 

Such,  however,  has  been  the  struggle  for  the  mastery  in  meta- 
physical reasoning,  with  a  view  to  establish  the  physical  laws  of 
the  universe  to  the  satisfaction  of  the  world,  as  to  have  enabled  cer- 
tain philosophers,  high  in  estimation  as  such,  to  play  off  before 
our  blunted  visions,  their  corpuscules  or  physical  atoms,  whereby 
they  would  attempt  to  explain  the  centre  of  gravity  between 
them,  the  curves  which  they  would  describe  while  revolving 
round  the  centre  of  gravity,  and  around  each  other,  &c. ;  and 
thence,  for  our  edification,  transfer  those  imagined  laws  to  the 
solar  system,  or  universe ;  and  so  ready  have  we  been  to  be 
satisfied  with  the  greatest,  or  most  approved  player,  that  opposi- 
tion has  given  greater  importance  and  impulse  to  the  Newtonian 
philosophy,  than  even  passive  assent  or  non-resistance.  And  it 
may  well  be  apprehended  that  it  was  more  from  some  knowl- 
edge of  the  fact,  than  from  inspiration,  that  Milton  was  enabled 
to  furnish  his  Raphael  with  so  appropriate  an  answer  to  Adam's 
inquiry  in  respect  to  the  movement  of  the  heavenly  bodies. 
But  a  fig  for  the  inspiration  of  a  poet  who  must  see  events  ful- 
filled before  he  can  sing  them. 

Notwithstanding,  it  is  properly  alleged,  and  generally  assent- 
ed  to,  that  "  curvature  is  a  change  of  direction  only ; "  neverthe- 


OP    THE    CIRCLE.  93 

less,  the  subject  has  been  much  perplexed  through  undue  or 
improper  considerations ;  and  hence  we  find  so  many  cases  put, 
the  declared  object  of  which  is  to  assist  our  conceptions  in 
respect  to  the  popular  doctrines  on  what  is  termed  a  delicate 
subject. 

Now  the  same  visionary  mode  of  reasoning,  (for  it  cannot  be 
philosophical,)  which  conceives  a  closer  and  closer  contact  in  the 
point  of  osculation  as  the  magnitude  of  the  circle  is  decreased, 
leads  to  the  idea  or  conception  of  curvilinear  angles  which  are 
incommensurable  and  incomparable ;  thus  an  osculation  is  said 
to  denote  an  assignable  angle  ;  and  hence  will  arise  that  un- 
meaning expression  of  an  angular  motion  of  a  heavenly  body ; 
and  although  the  circle  is  properly  named  the  measure  of  all 
angles,  nevertheless,  neither  the  periphery  of  the  circle,  nor  of 
any  curve  line,  is  supplied  with  angles,  nor  can  they  properly  be 
said  to  form  angles  with  their  tangents,  or  greater  or  less  angles 
with  their  tangents. 

As  is  properly  said,  curvature  is  a  change  of  direction  alone. 
This  change  of  direction  may  be  either  greater  or  less ;  and 
hence  may  be  said  to  denote,  indicate,  or  determine,  what  may 
be  called  the  curvature  or  convergency ;  and  if  referred  to  unity, 
it  will  indicate  what  may  properly  be  called  the  rate  of  curva- 
ture ;  or,  as  I  have  chosen  to  call  it,  while  treating  of  the  laws 
of  force  and  motion  of  the  heavenly  bodies,  the  rate  of  conver- 
gency, which  rate  of  convergency  I  proportion  to  that  of  the  rate 
of  force  or  gravity  by  which  it  is  produced,  very  differently  from 
Sir  Isaac  Newton. 

Now  it  would  seem  if  what  may  be  called  curvature,  or  con- 
vergency, is  an  element  for  consideration,  in  an  investigation  of 
the  laws  of  planetary  force  and  motion,  that  the  same  standard 
should  be  assumed  by  which  its  rate  may  be  measured  and  corn- 
pared  ;  and  such  standard  I  have  placed  in  unity;  nevertheless, 
we  find  the  most  approved  Newtonian  philosophers  and  com- 
mentators, —  while  treating  of  what  is  esteemed  so  very  impor- 
tant a  subject  as  that  of  the  Newtonian  department  of  the  invo- 
lution and  evolution  of  curves,  —  treating  the  subject  after  this 
manner : 

"  We  speak  of  curvature  that  is  greater  or  less ;  and  every 
person  has  a  general  knowledge  or  conception  of  the  difference, 
and  will  say,  that  an  ellipse  is  more  curved  at  the  extremities  of 
the  transverse  axis,  than  any  where  else.  But  before  we  can 
institute  a  comparison  between  them  with  precision,  &c.,  we 
must  agree  about  a  measure  of  curvature,  and  say  what  it  is 
we  mean  by  a  double  or  a  triple  curvature. 

"  Now  there'  are  two  ways  in  which  we  may  consider  curva- 


94  ON    THE    QUADRATURE 

ture  as  a  want  of  rectitude  ;  we  may  call  that  a  double  curvature 
which,  in  a  given  space,  carries  us  twice  as  far  from  a  straight 
line ;  or  we  may  call  that  a  double  curvature  by  which  we  devi- 
ate twice  as  much  from  the  same  direction.  Both  of  these 
measures  have  been  adopted ;  and  if  we  would  rigidly  adhere  to 
them,  there  would  be  no  room  for  complaint ;  but  mathemati- 
cians have  not  been  steady  in  this  respect,  and  by  mixing  and 
confounding  these  measures  have  frequently  puzzled  their  read- 
ers. All  agree,  however,  in  their  first  and  simple  measures  of 
curvature,  and  say,  that  the  curvature  of  an  arch  of  a  circle  is  as 
the  arch  directly,  and  as  the  radius  inversely.  This  is  plainly 
measuring  curvature  by  the  deflection  from  the  first  direction. 
In  an  arch  an  inch  long,  there  is  twice  as  much  deflection  from 
the  first  direction  as  when  the  radius  of  the  circle  is  of  half  the 
length.  If  the  radius  be  about  57 J  inches,  an  arch  one  inch  in 
length  will  produce  a  final  direction  one  degree  different  from 
the  first.  If  the  radius  be  114J  inches,  the  deviation  will  be  but 
half  a  degree.  The  linear  direction  from  the  straight  path  will 
also  be  one  half.  In  the  case  of  circles,  therefore,  both  measures 
will  agree ;  but  in  by  far  the  greatest  number  of  cases  they  may 
differ  exceedingly,  and  the  change  of  direction  may  be  greatest 
when  the  linear  deviation  is  least. 

"  Flexure,  or  change  of  direction,  is,  in  general,  the  most  sen- 
sible and  the  most  important  character  of  curvature,  and  is 
Understood  to  be  its  criterion  in  all  cases.  But  our  processes 
for  discovering  its  quantity  are  generally  by  first  discovering  the 
linear  deviation  ;  and  in  many  cases,  particularly  in  our  philo- 
sophical inquiries,  this  linear  deviation  is  our  principal  object. 
Hence  it  has  happened  that  the  mathematician  has  frequently 
stopped  short  at  this  result,  and  has  adapted  his  theorem  chiefly 
to  this  determination. 

"  These  differences  of  object  have  caused  great  confusion  in 
the  methods  of  considering  curvature,  and  led  to  many  disputes 
about  its  nature,  and  about  the  angle  of  contact ;  to  which  dis- 
putes there  will  be  no  end,  till  mathematicians  have  agreed  in 
their  manner  of  expressing  the  measures  of  curvature.  At  pres- 
ent we  abide  by  the  measure  already  given,  and  we  mean  to 
express  by  curvature  or  flexion  the  change  of  direction." 

This,  however,  does  not  indicate  any  great,  unity  of  purpose 
in  the  consideration  of  curvature  or  convergency ;  nor  could  the 
mathematicians  well  settle  the  question  by  convention,  so  as  to 
avoid  discrepancies  in  the  application. 

Now  the  same  author  above  quoted  says,  that  "  the  doctrine  of 
curves  by  involution  and  evolution  is  peculiarly  valuable  to  the 
speculator  in  the  higher  mechanics.  The  intensity  of  a  deflecting 


OF    THE    CIRCLE.  95 

force  is  estimated  by  the  curvature  which  it  induces  on  any  rec- 
tilineal motion  ;  and  the  variations  of  this  intensity,  which  is 
the  characteristic  of  the  force,  or  what  we  call  its  nature,  is  in- 
ferred from  the  variations  of  this  curvature." 

And  the  inference  which  they  have  drawn  is  properly  referred 
to  an  examination  of  the  Newtonian  method  of  investigating 
the  laws  of  planetary  force  and  motion,  in  which  some  proper 
estimation  is  attempted  to  be  given  in  respect  to  the  curvature 
or  convergency  produced  in  a  given  time  at  a  given  distance,  &c. 

That  portion  of  the  science,  however,  of  the  involution  and 
evolution  of  curves,  which,  from  its  supposed  transcendentalism, 
is  so  strenuously  claimed  by  Newton's  friends  and  admirers,  to 
be  wholly  of  Newtonian  origin,  seems  to  have  been  left  somewhat 
perplexed,  both  as  to  enunciation,  demonstration,  and  applica- 
tion ;  for  there  may  yet  be  some  doubt  in  respect  to  the  magnitude 
of  the  equi-curve  circle,  or  rather  in  regard  to  the  point  in  which 
it  osculates  the  curve ;  for  Newton  himself  perceived  the  closer 
and  closer  contact  as  epicircles  are  drawn  of  less  and  less 
magnitude,  ad  infinitum;  and  inasmuch  as  a  mere  point  without 
length  or  breadth  is  incapable  of  being  curved,  —  it  may  yet  be 
the  case  that  the  only  method  in  which  the  equi-curve  circle  can 
be  settled  and  established,  is  by  convention  ;  and  more  especially, 
inasmuch  as  the  osculating  circle  (as  named  by  Liebnitz)  forms 
an  actual  or  assignable  angle  with  the  curve  supposed  to  be 
measured.  By  such  means,  then,  they  obtain  their  fluent  for 
evolving  the  curve,  when  it  is  acknowledged,  that  after  exhaust- 
ing every  device,  no  fluent  could  be  found  for  evolving  the  circle. 
This  portion  of  the  science,  however,  is  not  without  its  use  to 
the  speculative  geometer  in  the  higher  mechanics,  as  it  enables 
the  expert  symbolical  analyst  to  equate  the  motions  of  the 
heavenly  bodies  to  their  crinkled  orbits,  even  though  they  assume 
the  force  of-  gravity  to  be  inversely  as  the  square  of  the  distance. 

Now  in  respect  to  the  simple  cycloid,  if  we  conceive  the  prime 
cylinder  (namely,  that  the  diameter  of  whose  inscribed  sphere  is 
unity)  to  be  rolled  upon  a  horizontal  plane,  so  as  to  perform  or 
describe  just  one  entire  revolution,  the  area  of  the  arch  formed 
by  that  line  of  the  cylinder  which  rests  upon  the  plane  at  the  time 
the  revolution  commences,  as  also  when  it  ends,  is  4,  or  ==.  the 
lateral  surface  of  the  circumscribed  cube  of  such  cylinder; 
while  the  base  of  such  arch  formed  on  the  plane  by  such  revolu- 
tion, is  =  the  lateral  surface  of  such  cylinder ;  or  is  =  the  entire 
surface  of  the  inscribed  sphere  of  such  cylinder.  It  is  also  =  the 
greatest  ellipse  or  elongated  circle  that  can  be  formed  in  the  recti- 
fied surface  of  such  cycloidal  arch ;  and  the  bulk  contained 
between  such  cyloidal  arch  and  its  base,  will  be  three  times  the 


96  ON    THE   QUADRATURE 

bulk  of  the  generating  cylinder.  The  area  of  the  greatest  par- 
abola that  can  be  drawn  within  such  arch,  is  =  four  times  the 
bulk  of  the  inscribed  sphere  of  the  generating  cylinder ;  or  is  = 
8-9ths  of  the  area  of  one  end  of  such  cycloiclal  arch,  or  is  to  the 
area  of  one  end  of  such  cycloiclal  arch,  as  two  thirds  of  1  is  to 
three  fourths  of  1.  But  the  subject  is  easy  of  contemplation. 

It  will  be  seen  from  the  gathered  hints  of  Sir  Richard  Phillips, 
that  Sir  Isaac  and  Sir  Richard  differ  very  widely  in  their  estima- 
tion of  the  fall  of  the  moon  from  a  tangent  in  one  minute  of 
time,  from  which  Sir  Isaac  deduced  his  proof  of  the  law  of 
gravity ;  Sir  Isaac  making  the  space  about  16  feet,  while  Sir 
Richard  makes  it  about  130,000  feet.  But  admitting  that  Sir 
Isaac  has  been  mistaken,  even  to  the  extent  alleged,  such  mis- 
take was  by  no  means  the  cause  of  the  tremendous  error  com- 
mitted by  him  in  respect  to  the  law  of  gravity. 

Perhaps  the  world  will  yet  come  to  the  conclusion  that  what 
is  termed  Newton's  first  law,  or  the  axiom  which  he  assumed  for 
the  government  of  the  world's  philosophy,  is  wholly  gratuitous 
and  uncalled  for,  being  merely  an  assumption  without  having 
any  known  foundation  in  the  universe ;  and  that  his  second  law 
or  axiom  should  at  least  have  stated  what  proportion  the  motive 
force  impressed,  beajs  to  the  change  of  motion  of  a  body. 
Nevertheless,  he  ought  to  have  been  convinced,  from  the  reason- 
ing which  naturally  flows  from  his  first  two  corollaries,  that  the 
motion  is  the  square  root  of  the  motive  force  applied ;  and  con- 
sequently, that  the  force  of  gravity  is  inversely  as  the  distance, 
and  not  as  the  square  of  the  distance.  But,  of  ail  the  discov- 
eries for  the  bewildering  and  misleading  of  philosophy,  perhaps 
nothing  could  have  been  contrived  that  could  so  perfectly  mysti- 
fy and  darken  counsel ;  that  could  so  effectually  direct  the  mind 
into  an  infinite  and  inexplicable  mysteriousness,  where  it  was 
supposed  ultimate  truths  alone  existed,  as  do  those  lemmas  upon 
which  his  mathematical  philosophy  chiefly  rests,  and  from  which 
those  refined  mathematics  have  emanated,  which  have  enabled 
La  Place  and  others  to  subdue  the  phenomena  of  the  heavens 
to  the  Newtonian  theory  and  law  of  gravity. 

Thus  the  mystical  terms  ultimate  ratios  and  ratios  of  equality, 
have  induced  the  belief  that  Sir  Isaac  Newton  found  that  the 
ultimate  ratios  of  the  sine,  chord,  and  tangent  of  arcs  infinitely 
diminished,  are  ratios  of  equality  ;  and  hence  that  in  our  reason- 
ing, we  may  use  the  one  for  the  other ;  and  having  once  com- 
menced such  a  train  of  reasoning,  we  must  come  to  the 
conclusion  that  the  ultimate  form  of  evanescent  triangles  formed 
by  the  arc,  chord,  and  tangent,  is  that  of  similitude,  and  their  ulti- 
mate ratio  is  that  of  equality,  and  that  each  may  be  used  for  the 


OF    THE    CIRCLE.  97 

other.  And  hence  has  been  deduced  that  futile  and  impractica- 
ble method  of  finding  the  length  of  the  arc  of  a  circle  in  terms 
of  the  sine,  versed  sine,  tangent  or  secant,  by  means  of  fluxions ; 
that  is,  in  geometrizing  upon  a  purely  geometric  point,  namely, 
upon  nothing,  or  mere  position ;  we  may  conceive  it  to  be  in 
what  shape  we  please  ;  that  its  ratios  are  ratios  of  equality,  and 
that  they  are  all  ultimate  ratios ;  and  hence,  that  the  evanescent  tri- 
angle may  be  in  form  of  the  circle,  and  vice  versa.  But  such 
has  been  the  philosophy  (if  we  may  so  speak)  which  has  led  the 
world  to  strange  conclusions. 

Now  it  is  said  that  Sir  Isaac  Newton  showed  that  the  space 
which  a  body  describes  by  any  finite  force,  whether  determinate, 
or  continually  varied,  are  to  each  other,  in  the  very  beginning  of 
the  motion,  in  the  duplicate  ratio  of  the  forces.  But  this  is  di- 
rectly deduced  from  the  simple  law  of  falling  bodies,  which  had 
long  before  been  determined  by  Galileo.  Nevertheless,  the  fact, 
however  disclosed,  should  have  taught  Newton,  or  any  other 
person,  that  the  force  of  gravity  is  not  inversely  as  the  square  of 
the  distance  ;  but  that  it  is  inversely  as  the  distance. 


13 


CHAPTER    II. 


On  the  Law  of  Gravity,  and  the  Popular  Deduction  and 
Promulgation  of  the  Supposed  Law. 

SECTION    FIRST. 

GRAVITY  may  properly  be  defined  to  be  (at  least  for  the  pur- 
poses of  this  article)  that  physical  power  which  pervades  the  solar 
system,  (and  perhaps,  judging  by  analogy,  the  universe,)  dis- 
played or  manifested  to  our  senses  in  and  through  the  medium 
of  tangible  matter,  by  which  the  phenomenon  is  constantly  pre- 
sented to  our  perception,  of  aggregate  bodies  of  matter  attracting 
each  other,  at  least  under  certain  circumstances  ;  and  hence  we 
observe  bodies  falling  from  a  state  of  rest  towards  the  centre  of 
the  earth  ;  and  hence  the  centripetal  force  which  retains  the 
planets  in  their  orbits  around  the  sun,  and  the  satellites  in  their 
orbits  around  their  primaries.  To  present  and  more  readily 
compare  the  popular  notions  of  gravity  with  suggestions  or  hy- 
potheses which  I  may  attempt  to  intrude  in  place  of  them,  I 
will  suggest  the  consideration  of  the  subject  under  two  heads, 
namely :  the  theory  of  gravity,  and  the  law  of  gravity.  The  the- 
ory of  gravity  will  mainly  be  referred  to  the  momentous  ques- 
tion, —  whether  tangible  or  aggregate  matter,  of  itself,  is  endued 
with  or  possessed  of  innate  or  inherent  gravity,  or  an  inherent 
power  to  attract  and  be  attracted,  Justin  proportion  to  the  quantity 
of  aggregate  matter,  wherever  situated,  in  the  solar  system,  (for  I 
will  not  attempt  to  investigate  beyond  that,)  which  theory  may  be 
termed  the  theory  of  inherent  gravity  ;  — or,  whether  the  power 
incident  to  a  given  quantity  of  matter  to  attract,  or  the  susceptibil- 
ity to  be  attracted,  must  depend  upon  some  numerical  law  in 
respect  to  its  distance  from  the  sun,  or  great  centre  of  gravity  of 
the  solar  system,  —  which  may  be  termed  incidental  or  incited 
gravity. 

The  law  of  gravity,  in  this  division  of  the  subject,  will  simply 
consist  in  the  rate,  intensity  or  efficacy  which  the  force  of  gravity 


ON    THE    LAW    OF    GRAVITY.  99 

exerts,  between  two  given  quantities  of  matter,  in  proportion  to 
the  distance  between  them  at  which  it  shall  operate ;  and  hence, 
to  determine  the  numerical  law  of  gravity,  is  simply  to  determine 
the  rate  or  intensity  in  proportion  to  distance.  And  in  the  in- 
vestigation of  such  ratio  and  proportion,  unity  or  1  occupies  a 
very  conspicuous  numerical  point  or  position.  If  the  theory  of 
inherent  gravity  shall  prevail,  it  will  be  necessary  to  consider  the 
theory  and  law  of  gravity  under  distinct  heads  ;  and  consequent- 
ly, that  the  theory  may  be  brought  to  harmonize  with  the  law  in 
some  of  the  principal  phenomena  produced,  some  sufficient 
cause  must  necessarily  be  assumed.  Hence  we  find  that  Sir 
Isaac  Newton  assumed  that  planets  at  a  greater  distance  from 
the  sun  were  less  dense  than  those  which  were  nearer ;  and  con- 
sequently that  their  powers  of  gravity  were  as  their  respective 
bulks  multiplied  into  their  rates  of  density,  or  as  their  quantity 
of  matter  in  proportion  to  their  bulk ;  but  as  the  supposed  densi- 
ties of  those  planets  having  no  satellites,  could  not  be  as  readily 
determined  and  proportioned  as  those  which  had  satellites,  resort 
was  had  to  their  supposed  disturbing  forces  operating  on  other 
planets ;  and  hence  their  supposed  densities  were  variously  as- 
sumed, according  to  the  supposed  evidence  obtained. 

So  nearly,  however,  have  such  men  as  Euler,  LaGrange  and 
others  found  their  supposed  proportional  densities  to  agree  with 
their  proportional  distances  from  the  sun,  (namely,  in  the  inverse 
ratio  to  that  of  distance,)  that  they,  and  especially  LaGrange, 
founded  astronomical  tables  upon  this  hypothesis  ;  and  long  af- 
ter, he  remarked  that  he  saw  no  occasion  to  vary  from  a  strict 
adherence  to  such  hypothesis.  •  To  be  sure,  Herschel,  the  astron- 
omer, assumed  the  density  of  the  planet  Herschel  to  be  greater 
than  that  of  Saturn  ;  but  modern  discoverers  have  determined  and 
shown  that  Herschel  was  mistaken  in  respect  to  the  motion  of 
Herschel's  satellites  ;  and  consequently  in  respect  to  the  supposed 
density  of  the  planet ;  and  in  regard  to  the  supposed  density  be- 
ing proportioned  to  the  distance,  as  with  other  planets;  and 
hence  upon  the  theory  of  inherent  gravity,  the  proportional  dens- 
ities of  the  planets  are  assumed  to  be  inversely  as  their  respec- 
tive distances  from  the  sun,  for  the  reason  that  the  power  of 
gravity  which  they  appear  to  exert  is  in  proportion  to  their  re- 
spective distances  from  the  sun,  or  the  centre  of  gravity  of  the 
solar  system.  But  may  it  not  well  be  doubted  whether  theon  ti- 
cal  or  hypothetical  philosophy  was  well  applied  in  attempting  to 
deduce  the  cause  from  the  effect,  —  from  the  facts  and  the  phe- 
nomena observed, — when  a  Want  of  density  in.  proportion  to  the 
bulk  was  assumed  as  the  cause  of  the  want  of  gravity  in  propor- 
tion to  the  bulk  ?  This  depends,  however,  upon  the  premises 


100  ON    THE    LAW    OF    GRAVITY. 

from  which  we  draw  our  inferences;  as  such  a  deduction  of  the 
cause  is  at  best  but  an  inference,  and  should  be  drawn  from  the 
best  and  greatest  amount  of  evidence  that  may  be  brought  into 
the  case,  and  should  not  be  left  to  rest  foe  its  truth  or  final  sup- 
port, upon  some  abstract  hypothesis  previously  assumed,  —  as 
for  instance,  the  innate  or  inherent  gravity  of  matter.  If  the  at- 
tractive power  of  matter  be  inherent  in  if,  and  that  too  in  pro- 
portion to  the  quantity,  wherever  situated  in  the  solar  system,  then, 
to  be  sure,  the  popular  inference,  —  that  the  proportional  densi- 
ties of  the  respective  planets  are  as  their  bulks  multiplied  into 
the  inverse  of  their  respective  distances  from  the  sun,  —  is  well 
founded,  because  their  proportional  attractive  powers  are  such  ; 
but  perhaps  if  the  naked  or  abstract  hypothesis  of  the  inherent 
gravity  of  matter  had  not  been  brought  in  as  a  make-weight,  phi- 
losophers would  not  have  seriously  thought  of  making  the  popular 
deduction,  or  of  drawing  the  popular  inference  in  respect  to  the  dif- 
ferent densities  of  the  planets,  especially  from  any  observed  facts 
or  phenomena,  but  would  have  stricken  from  the  declaration 
such  allegations  as  were  not  proved,  and  not  founded  upon  the 
evidence ;  that  is,  they  would  have  travelled  upon  facts,  so  far  as 
they  were  plainly  to  be  discovered,  and  let  the  inferences  or  de- 
ductions follow  in  their  proper  places  ;  and  instead  of  alleging 
that  the  proportional  densities  of  the  planets  are  respectively  as 
the  product  of  the  bulk  X  the  inverse  of  the  distance,  they  would 
have  said  that  the  proportional  power  of  gravity  exercised  in,  by, 
or  through  the  respective  planets,  is  as  the  product  of  the  bulk  X 
by  the  inverse  of  the  distance,  (for  here  is  a  fact  found,)  and 
would  have  left  the  cause  (if  not  yet  discovered  through  the  me- 
dium of  facts  or  phenomena)  to  be  assigned  whenever  warrant- 
ed, instead  of  assuming  a  naked,  unsustained  allegation,  as  the 
cause  of  the  phenomena  observed. 

Let  the  inquiring  mind,  then,  conceive,  in  accordance  with 
the  popular  notion  or  hypothesis,  that  the  planets  become  less 
and  less  dense  in  proportion  to  their  distance  from  the  sun,  and 
it  will  inquire  for  the  cause  of  this  very  curious  phenomenon,  — 
this  phenomenon  apparently  fraught  with  some  unity  of  purpose, 
—  some  indicative  of  some  necessary  law,  by  which  the  solar 
system  (if  not  the  universe)  is  governed  ;  and  it  would  be  slow 
to  believe  that  all  this  order  and  harmony  is  merely  fortuitous, 
or  the  result  of  chance ;  nor  would  it  be  very  likely,  on  serious 
reflection,  to  base  or  assign  the  cause  in  a  naked  and  wholly  un- 
supported assumption  ;  and  especially  if  such  assumption  were 
in  direct  derogation  of  all  the  physical  laws  and  operations  of 
the  universe,  so  far  as  we  are  able  to  enter  into  their  examina- 
tion and  their  operations,  by  means  of  such  puny  attempts  and 


ON    THE    LAW    OF    GRAVITY.  101 

trials  as  we  are  enabled  to  make.  Neither  would  the  inquiring 
mind,  —  with  the  view  to  cut  short  its  inquiries,  and  dispose  in- 
stanter  of  a  question  or  hypothesis  in  reference  to  which  the  uni- 
verse is  to  be  squared  or  quadrated,  and  upon  which  the  whole 
superstructure  of  astronomy  is  to  be  reared,  — be  likely  to  adopt 
as  the  cause  of  the  most  astonishing  order  and  harmony  in  the 
solar  system,  that  naked  hypothesis  which  makes  the  whole 
physical  power  of  the  universe,  — a  power  to  be  constantly  exer- 
cised at  immense  distances,  —  to  be  inherent  in  inert  or  dead 
matter.  If  it  be  possible  to  have  a  clear  conception  of  such  a 
matter,  it  is  but  conceiving  a  state  of  things  existing  per  se,  or,  as 
some  have  imagined  the  case  to  have  been,  when  the  earth  was 
without  form  and  void.  And  will  any  one  pretend  that  such 
hypothesis  is  any  improvement  upon  the  heathen  mythology, 
which  places  fate  beyond  the  power  of  the  gods  to  control  it  ?  I 
say,  the  inquiring  mind  will  hesitate  before  it  adopts  as  the 
basis  of  the  whole  fabric  of  astronomy  and  philosophy,  a  naked 
allegation  or  hypothesis,  which,  if  true,  so  fully  dispenses  with 
any  law  of  order  by  which  we  hope  and  believe  the  universe 
is  upheld,  controlled  and  constantly  governed,  and  converts  the 
whole  into  a  system  (if  such  it  may  be  called)  of  disturbing 
forces, —  that  is,  a  system  which  constantly  disturbs  itself  into  an 
eternal  harmony.  Unless,  however,  it  had  been  believed,  (as 
many  good  and  eminent  men  have  believed  implicitly,  even  to 
the  endorsement  of  the  idea  from  the  pulpit,)  that  the  projectile 
hypothesis  of  Sir  Isaac  Newton,  —  namely,  that  the  planets 
were  originally  projected  by  the  Supreme  Being  in  tangents  to 
some  given  point  of  their  respective  orbits,  —  was  a  sufficient 
development  of  a  Deity,  and  of  his  doings  in  upholding  and 
governing  the  solar  system,  and,  by  analogy,  the  universe,  — 
such  hypothesis  could  not  have  prevailed.  But  I  dare  not  so 
rashly  presume  or  assume,  but  if  I  shall  be  able  satisfactorily 
to  determine  something  in  respect  to  a  numerical  development 
of  the  great  physical  law  by  which  the  solar  system  is  upheld, 
sustained  and  governed,  I  shall  be  satisfied  to  dispense  with 
mere  naked  and  presumptuous  allegations,  which,  if  issuing 
from  a  man  proudly  sustained,  as  was  Sir  Isaac  Newton,  might 
plunge  the  world  into  errors  from  which  centuries  could  not 
extricate  it. 

The  true  theory  of  gravity,  then,  is  yet  open  to  investiga- 
tion ;  and  whether  the  force  of  gravity  be  a  power  inherent 
in  matter,  or  whether  gravity  or  attraction  be  an  omnipresent  law, 
operating  on  matter  in  proportion  to  distance  from  a  centre  of 
gravity,  —  is  a  question  which  ought  to  be  determined  upon  the 
best  evidence  the  nature  of  the  case  will  afford.  If,  however,  it 


102  ON    THE    LAW    OP    GRAVITY. 

should  be  discovered  that  the  attractive  power  of  a  body  of  ag- 
gregate matter  in  the  solar  system  is  only  induced,  and  that,  too, 
in  proportion  to  its  inverse  distance  from  the  sun,  or  centre  of 
gravity  of  a  system,  the  theory  of  gravity  will  become  so  far  in- 
volved in  a  consideration  of  the  law  of  gravity,  as  to  require  no 
distinct  or  separate  consideration  from  lhat  of  the  law. 

In  the  consideration  or  investigation  of  this  subject,  it  will  be 
my  object  to  bestow  most  attention  upon  the  law  of  gravity ; 
namely,  its  rate  or  intensity  as  proportioned  to  the  distance  from 
what  may  be  properly  termed  the  attracting  body,  or  that  which 
induces,  or  from  which  appears  to  emanate  the  centripetal  force 
which  retains  a  revolving  body  in  its  orbit ;  treating,  however,  of 
the  Newtonian  theory  of  gravity  in  an  incidental  manner,  when- 
ever it  may  seem  necessary  by  way  of  explication.  It  may  here 
be  remarked  that  the  Newtonian  method  of  investigating  the  law 
of  gravity  by  a  consideration  of  the  law  of  falling  bodies  towards 
the  earth,  has  been  preserved  with  much  care  by  those  authors 
who  have  quoted  or  given  us  the  method,  and  who  have  com- 
mented on  it.  See  Maclaurin's  account  of  the  Newtonian  philoso- 
phy, in  four  books,  Rees'  Cyclopaedia,  Vince's  Universal  System 
of  Physical  Astronomy,  &c. ;  the  investigation  seems  to  have 
been  extremely  simple  in  its  character,  and,  from  that  cause  alone, 
would  not  seem  calculated  either  to  lead  mankind  into  error,  or 
to  keep  them  there,  if  peradventure  they  were  thus  led ;  for  the 
investigation  certainly  contains  nothing  more  intricate  than 
the  consideration  of  a  numerical  formula,  designed  to  explain 
the  law  of  falling  bodies.  The  simplicity  of  the  method,  how- 
ever, does  not  deny  to  it  the  susceptibility  of  affording  ample 
demonstration  of  the  law  of  gravity ;  namely,  satisfactory  and  in- 
dubitable proof  to  the  mind,  in  respect  to  the  rate  or  intensity  of 
gravity  as  proportioned  to  distance  ;  and  whenever  such  indubi- 
table proof  is  understandingly  presented  to  the  mind,  whether  by 
a  numerical,  a  mathematical,  or  a  geometrical  process,  if  so  be 
that  ratio  is  the  talisman  that  furnishes  the  proof,  such  proof  may 
properly  be  called  a  demonstration.  I  speak  thus  with  a  view  to 
the  extreme  simplicity  in  which  demonstrations  upon  the  most 
important  points  relative  to  the  science  of  astronomy  may  be 
presented,  and  especially  those  relating  to  fundamental  princi- 
ples upon  which  the  superstructure  should  be  reared.  Such  is 
the  case  in  regard  to  the  law  of  falling  bodies  towards  the  earth, 
— from  which  it  has  been  supposed  that  the  law  of  gravity  has 
been  deduced ;  which  law  of  falling  bodies,  and  the  consequent 
deduction  of  the  law  of  gravity,  depend  upon  a  proper  exposition 
of  one  of  the  most  simple  numerical  formulas  that  can  be  devised, 
although  the  law  of  gravity  may  not  be  as  simply  or  as  beauti- 


ON    THE    LAW    OF    GRAVITY.  103 

fully  developed  and  presented  to  the  mind,  by  a  consideration  of 
the  law  of  falling  bodies,  as  by  a  consideration  of  Kepler's  laws, 
(leaving  out  his  elliptical  hypothesis.)  Nevertheless,  as  the 
method  pursued  for  obtaining  a  knowledge  of  the  law  of  gravity 
has  been,  by  a  consideration  of  the  law  of  falling  bodies  ;  and  as 
a  proper  consideration  of  the  same  will  deduce  a  proper  knowl- 
edge of  the  law  of  gravity  ;  the  labor  bestowed  upon  the  subject 
will  be  directed  mostly  to  a  deduction  of  the  law  of  gravity  from 
a  consideration  of  the  law  of  falling  bodies ;  as  thereby  the  pop- 
ular methods  may  be  quoted,  and  the  popular  modes  or  manner 
of  investigation  compared  with  my  own. 

Nor  is  truth  or  demonstration  the  less  valuable  in  consequence 
of  its  simplicity,  even  though  it  may  lie  at  the  very  foundation  of 
the  science  of  astronomy ;  for  if  it  were  so,  the  truths  disclosed 
by  Kepler  would  be  nearly  valueless ;  for  a  more  simple  disclo- 
sure of  the  most  important  truths  could  scarcely  be  made,  —  a 
disclosure,  too,  developed  and  explained  by  the  aid  of  our  com- 
mon numerical  or  Arabic  characters,  but  from  which,  even  yet, 
all  physical  truths  worthy  of  the  science  of  astronomy,  have  em- 
anated. 

Let  the  world,  then,  understand,  —  let  mankind,  who  have 
supposed  demonstrated  truths  to  be  beyond  their  comprehension, 
understand,  —  that  the  value  of  truth  consists  in  the  benefit  which 
it  bestows,  —  not  in  its  complexity ;  and  that  demonstrable  truths 
are  vastly  nearer  to  their  comprehension  than  many  have  been 
led  to  believe.  But  notwithstanding  the  extreme  simplicity  and 
ease  of  comprehension  of  the  demonstrations  of  the  most  impor- 
tant truths,  (and  especially  in  relation  to  the  science  of  astron- 
omy,) care  should  be  taken  that  we  do  not  extend  the  applica- 
tion of  a  demonstration  beyond  its  legitimate  province,  lest  we 
thereby  deduce  an  inference  which  has  no  foundation  in  truth  ; 
which  inference,  when  once  engrafted  into  the  science,  is  thence- 
forward implicitly  received  as  truth.  And  such  I  have  supposed 
to  be  the  case  in  respect  to  the  supposed  demonstration,  by  Sir 
Isaac  Newton,  of  Kepler's  elliptical  hypothesis,  which  has  con- 
tributed so  much  to  the  fame  of  Sir  Isaac  Newton. 

Kepler,  in  making  calculations  upon  observations  previously 
made  by  Tycho,  (in  which  observations  Kepler  had  great  confi- 
dence,) came  to  the  conclusion  that  the  orbit  of  Mars  was  an  el- 
lipse,—  or  an  oval,  as  he  called  it,  —  and  he  afterwards  came  to 
the  conclusion  that  the  orbits  of  all  the  planets  were  elliptical ; 
and  he  found  from  observation  that  the  motions  of  planets,  at 
perihelion  and  aphelion,  varied  inversely  as  their  distances  from 
the  sun  varied ;  but  he  was  unable  to  determine  whether  this  was 
a  general  law  throughout  the  orbit,  but  he  supposed  it  to  be ;  and 


104  ON    THE    LAW   OF    GRAVITY. 

if  so,  the  radius  vector  would  always  describe  equal  areas  in  equal 
times. 

Newton  applied  that  proposition  of  Euclid  which  teaches  that 
"  triangles  upon  equal  bases,  and  between  the  same  parallels  or 
equal  altitudes,  are  equal  to  one  another,"  which  fully  confirmed 
the  truth  of  what  Kepler  had  suggested,  that  the  variation  of  the 
motion  is  in  the  inverse  ratio  to  the  variation  of  the  distance  ; 
and  that  too  in  all  parts  of  the  orbit ;  and  consequently  that  the 
radius  vector  would  always  pass  over  equal  areas  in  equal 
times.  But  that  such  demonstration  goes  to  the  extent  of  estab- 
lishing Kepler's  suggestions  of  elliptical  orbits,  perhaps  none,  on 
reflection,  will  seriously  contend.  The  demonstration  itself  is 
only  applicable  to  one  of  the  ascertained  laws  of  Kepler ;  namely, 
a  law  of  motion  depending  for  its  phenomena  upon  the  great 
physical  law  of  gravity,  and  not  at  all  upon  his  elliptical  hypothe- 
sis ;  for  it  will  at  once  be  seen  that  such  demonstration  applies 
as  well  to  a  circular  orbit,  whether  centric  or  eccentric,  as  to 
an  ellipse ;  and  that  it  will  even  apply  to  an  orbit  composed  of  a 
polygon  of  rectilinear  sides,  so  long  as  the  planet  shall  be  per- 
forming one  side  of  the  polygon  ;  for  such  is  the  condition  upon 
which  the  proposition  holds  true,  or  upon  which  geometry  is  a 
science  to  be  relied  upon. 

Such  demonstration,  then,  has  nothing  whatever  to  do  with 
the  elliptical  hypothesis,  and  should  not  have  been  used  in  its 
support. 

SECTION    SECOND. 

It  is  possible,  that  from  the  days  of  Aristotle  to  the  present, 
Physics  and  Metaphysics  (especially  so  far  as  dialectics,  or  the 
proper  modes  of  reasoning,  are  concerned)  have  been  treated  of 
with  a  distinction  not  wholly  consonant  with  the  best  philosophy. 
Those  barriers,  however,  have  been  somewhat  broken  down  by 
some  modern  philosophers  and  profound  thinkers,  among  whom 
I  will  mention,  by  way  of  eminent  example,  Kepler  and  Kant, 
the  philosophy  of  whom,  taken  in  connection,  and  carefully  an- 
alyzed and  compared,  would  form  the  basis  of  a  connected  phi- 
losophy, which,  if  the  true  intent  and  meaning  were  not  frittered 
away  by  false  pretenders,  or  obscured  by  an  occult  philosophy 
which  no  mortal  can  comprehend,  would  not  only  become  ac- 
cessible to  a  vast  portion  of  mankind  who  now  grope  in  the 
darkness  of  philosophic  error,  in  which  it  has  been  shrouded  by 
many  who  claim  to  dispense  science  to  the  world,  but  would  be 
worthy  of  that  dignity  of  man  to  which  science  alone  seems  des- 
tined to  elevate  him.  If,  however,  either  of  those  champions  in 
philosophy  should,  from  too  abstract  a  consideration  of  the  sub- 


ON    THE    LAW    OP    GRAVITY.  105 

ject  on  which  they  may  be  particularly  treating,  deviate  in  any 
respect  from  those  dialectic  principles  upon  which  all  true  philos- 
ophy should  be  based,  and  which  is  not  properly  the  result  of 
human  logic,  I  nevertheless  claim  for  mankind  the  right,  the 
privilege  to  pause  for  consideration,  rather  than  implicitly  follow 
any  file-leader  to  inexplicable  conclusions. 

Hence  the  monadology  of  Liebnitz  with  the  incomprehensible 
consequences  to  which  he  would  extend  the  hypothesis,  however 
beneficial  such  hypothesis  may  be  in  disciplining  the  mind  to  a 
remarkable  degree  of  astuteness,  may,  however,  savor  somewhat 
of  that  mysticism  with  which  the  Pythagorean  theory  of  monads 
and  dyads  has  been  accused ;  there  is  not,  perhaps,  any  philos- 
ophical objection  to  either  Kepler,  Kant,  or  Liebnitz,  so  far 
as  they  have  been  actuated  in  their  researches  by  the  Pytha- 
gorean philosophy,  or  manner  of  investigation,  —  notwith- 
standing certain  conclusions  or  consequences  may  be  imag- 
ined to  result,  which  transcend  the  bounds  of  absolute  investiga- 
tion. It  is  said  that  the  allegation  of  Pythagoras  in  respect 
to  his  philosophy,  was  conclusive  ;  insomuch,  that  the  phrase 
"  he  said  so  "  was  sufficient  authority  without  any  proof.  But  it 
is  believed  that  the  German  philosophers,  to  whom  I  have  re- 
ferred, never  aspired  to  be  more  than  mortals  while  living,  or  to 
have  divine  honors  paid  them  after  death  ;  this  kind  of  ambition 
being  incident  to  another  person  and  nation  ;  and  hence,  in  what 
are  termed  our  highest  and  best  works  on  astronomy,  the  whole 
proof  adduced  in  support  of  most  material  matters  of  which  we 
should  like  to  know,  or  be  informed,  often  consists  in  that  ex- 
pression so  obnoxious  to  science,  that  "  Sir  Isaac  Newton 
found,"  &c. ;  and  it  is  somewhat  surprising  that,  in  this  liberal 
age  of  inquiry,  even  the  learned  should  be  more  willing  to  rely 
on  such  implicit  faith,  than  to  take  the  trouble  to  examine  into 
the  truth  or  error  couched  under  the  naked  allegation.  But  it  is 
yet  to  be  hoped  that,  on  serious  consideration,  infallibility  will 
be  considered  an  attribute  which  no  mere  mortal  should  impute 
to  himself;  and  more  especially,  that  the  true  dignity  of  man 
can  never  condescend  to  award  such  a  meed  of  praise  to  a  mer 
fellow  mortal. 

If  Kepler,  after  an  analogical  course  of  reasoning  upon  the 
eternal  fitness  of  things,  by  which  he  deduced  certain  physical 
laws,  more  beneficial  to  science  than  ever  fell  to  the  lot  of  any 
other  mortal,  eventuatly  fell  into  an  absurd  conclusion,  from 
placing  too  much  confidence  in  the  astronomical  observations  of 
Tycho,  it  is  certainly  no  more  than  justice  to  him  and  to  the 
world,  to  expose  and  correct  such  error.  If  Leibnitz  became  alto- 
gether too  sensitive  at  losing  the  credit  of  what  he  esteemed  his 
14  ' 


106  ON    THE    LAW    OF    GRAVITY. 

own  discoveries  in  mathematics,  it  should  be  recollected  that 
Mercator  but  just  escaped  a  similar  fate,  and  that  poor  Flamstead 
was  totally  ruined  by  the  same  means.  And  it  may  be  quite 
doubtful,  were  Liebnitz  to  burst  the  cerement  which  shrouds 
him,  and  to  come  forth  at  this  period  of  time,  and  behold  the  ap- 
plication of  his  discovery  to  what  is,  in  the  most  emphatic  sense, 
termed  "  the  Newtonian  System,"  (viz.  its  application  to  the 
Newtonian  theory  and  law  of  gravity,)  whether  he  would  not 
wish  himself  wholly  freed  from  even  the  suspicion  of  having  been 
the  discoverer.  If,  however,  his  supposed  discovery  of  an  infinite 
series  for  the  surface  of  the  circle,  is  founded  upon  those  errors 
which  were  and  still  are  extant,  and  consequently  lead  to  erro- 
neous results,  his  fame,  like  that  of  any  other  mortal,  should,  nev- 
ertheless, stand  upon  its  true  merits,  viz.  that  of  the  development 
of  beneficial  truths.  Kant,  with  all  his  profundity  of  knowledge, 
doubtless  applied  his  mind  much  more  intensely,  and  reasoned 
much  clearer  upon  that  compartment  of  philosophy  which  has  in 
general  been  termed  metaphysics,  than  upon  what  is  generally  call- 
ed physics ;  hence,  in  the  analyzation  of  the  human  mind,  he  was 
doubtless  far  more  astute  and  better  versed,  than  in  those  merely 
physical  sciences,  the  apparent  truths  of  which  are  but  the  results 
of  the  mind ;  and  although  he  seemed  well  to  understand  the 
machinery  of  the  human  mind,  by  which  a  knowledge  of  physi- 
cal science  is  developed,  nevertheless,  inasmuch  as  physical  re- 
sults were  but  secondary  objects  in  his  Critique  of  Pure  Reason, 
and  having  written  his  Natural  History  of  the  Theory  of  the 
Heavens  according  to  the  Newtonian  System,  previous  to  his 
closer  application  to  an  analysis  of  the  human  mind,  there,  per- 
haps, can  be  no  good  reason  why  he  might  not  have  fallen  into 
the  same  popular  errors  with  others,  in  respect  to  the  Newtonian 
theory,  inasmuch  as  the  subject  was  popular,  and  generally  re- 
ceived as  tantamount  to  demonstrated  truth. 

But  after  having  so  clearly  and  conclusively  established  his 
categories  in  respect  to  what  forms  the  pure  intuitions  of  the  hu- 
man mind,  —  viz.  space  and  time,  —  as  also  those  which  are  indis- 
pensable to  our  understanding,  from  which  results  and  conclusions 
are  deduced,  it  would  seem  almost  incredible  that  a  mind  like 
his,  with  the  elements  of  time  and  space  so  well  considered  and 
understood,  should  not  have  discovered  the  most  important  errors 
in  the  Newtonian  theory  and  law  of  gravity ;  and  more  especial- 
ly, as  the  great  law  of  Kepler,  that  the  "  squares  of  the  periods 
of  the  planets  are  as  the  cubes  of  their  mean  distances  from  the 
sun,"  is  the  most  lucid  exposition  or  commentary  .of  his  category 
in  respect  to  time  and  space,  that  can  well  be  conceived.  In 
the  Keplerean  laws,  we  find  mathematical  results  corresponding 


ON    THE    LAW    OF    GRAVITY.  107 

with  and  proportioned  to  time  and  space,  however  varied  such 
time  and  space  may  be ;  and  for  this  very  reason,  it  is  perhaps 
the  best  corroborative  evidence  extant,  in  favor  of  a  true  devel- 
opment of  the  quadrature  of  the  circle ;  inasmuch  as  an  abstract 
mathematical  consideration  of  the  circle  is  destitute  of  time,  as  a 
comparative  element.  And  hence  I  will  again  suggest,  that  per- 
adventure,  physics  and  metaphysics,  have,  in  general,  been  con- 
sidered in  too  distinctive  a  manner ;  for  if,  as  taught  by  Kant,  we 
cannot  perceive  anything  except  by  means  of  certain  original, 
necessary,  unchangeable  forms  of  thought ;  —  if,  to  produce  re- 
sults, the  categories  must  be  applied  to  external  objects,  by  which 
application  they  only  become  subject  to  error ;  if  the  demonstra- 
tive certainty  of  mathematics,  the  objects  of  which  are  space, 
time,  form,  &c.,  or  whatever  quantity  may  be  numerically  ex- 
pressed and  proportioned,  lies  in  the  necessity  of  the  forms  of 
thought,  and  not  in  the  range  of  error  to  which  experience  is 
subject ;  surely  the  most  intimate  connection  exists  between  the 
physical  and  metaphysical  world,  so  far  at  least  as  the  human 
mind  is  concerned ;  nor  could  we,  perhaps,  readily  distinguish 
this  intimate  connection  from  that  of  a  unity  of  purpose  in  Him 
who  created  all  things.  If  it  be  then  the  object  of  the  Critique 
of  Pure  Reason,  to  mark  out  and  define  the  categories,  or  tran- 
scendental ideas  or  pure  intuitions,  or  the  necessary  and  un- 
changeable forms  of  thought,  and  to  direct  their  proper  applica- 
tion to  external  objects  with  the  view  to  avoid  error  in  our  results, 
the  table  of  categories  should  not  only  be  full  and  ample,  but 
most  explicitly  explained  or  defined;  and  although  they  may  all 
be  comprehended  in  the  four  classes  to  which  they  are  assigned 
by  Kant,  viz.,  quantity,  quality,  relation  and  modality,  it  may  per- 
haps be  questioned,  whether  Kant's  division  of  the  first  class,  or 
rather  whether  those  categories  assigned  to  the  first  class, —  viz. 
unity,  multitude  and  totality,  —  give  a  full  and  proper  intimation 
of  all  that  would  seem  to  be  comprehended  under  that  head. 
To  be  sure,  unity,  multitude,  and  totality,  are  so  far  transcenden- 
tal or  intuitive,  as  not  to  depend  on  experience,  or  be  subject  to 
error  from  a  misdirected  course  of  reasoning ;  but  when  applied 
to  those  passive  qualities  or  elements,  in  which  all  operations 
exist  and  take  place,  —  viz.,  space  and  time,  —  which  are  as  appli- 
cable to  quantity,  numerically  expressed,  as  are  the  elements  of 
force  or  motion,  or  even  a^fy  tangible  form  whatever,  —  and  in- 
asmuch as  quantity  (however  great  or  small)  may  always  be 
taken  or  assigned  at  unity,  and  consequently,  quantity,  however 
great  or  small,  may  emphatically  be  the  monad,,  or  unity  of  Py- 
thagoras, and  the  source  of  all  numbers,  or  more  properly  the 
source  of  all  ratio  and  proportion,  (for  I  will  not  wholly  discard  the 


108  ON    THE    LAW    OP    GRAVITY. 

Pythagorean  philosophy  of  numbers,  from  which  our  happiest  and 
mosl  beneficial  results  have  been  derived,)  —  it  would  seem  as 
though  what  Kant  has  seen  proper  to  call  unity,  might  have  been 
denoted  by  the  term  quantity,  in  which  case,  its  adjuncts  would 
have  been  denoted  by  those  very  expressions  from  which  ratio  and 
proportion  (those  indispensable  elements  in  human  wisdom)  ema- 
nate or  are  deduced.  Thus  the  adjuncts  would  have  been  a  greater 
or  a  less  quantity,  or  an  increase  or  decrease  of  quantity,  &c.,  and 
I  am  unable  to  comprehend,  that  when  applied  to  space  or  lime, 
the  idea  of  more  or  less,  or  greater  and  less,  is  less  transcen- 
dental or  intuitive,  or  that  it  is  a  less  original  or  necessary 
form  of  thought,  than  that  of  multitude  or  totality;  and  if  so,  the 
former  certainly  would  seem  far  better  adapted  to  a  proper  applica- 
tion of  the  understanding,  a  priori,  to  external  objects  or  objects  of 
research.  If,  then,  the  mind  cannot  have  any  real  conception  of 
quantity  in  the  abstract,  without  having  at  the  same  lime,  a  con- 
ception of  greater  or  less  quantity,  I  would  certainly  supply  the 
place  of  unity,  multitude  and  totality,  by  quantity,  a  greater 
quantity,  and  a  less  quantity.  If  quantity  denoted  by  unity,  or 
assumed  at  unity,  be  an  abstract  idea,  (and  perhaps  abstractly, 
quantity  cannot  be  considered  otherwise;)  and  if  quantity,  con- 
sidered in  the  abstract,  be  numerically  accompanied  with,  or  be 
contained  in  the  idea  of  more  and  less  quantity,  that  is,  that 
unity,  however  great  or  small,  may  be  either  increased  or  dimin- 
ished;  then,  to  be  sure,  that  indispensable  element  to  human 
wisdom  in  the  investigation  of  all  phenomena,  viz.  that  of  ratio 
or  proportion,  lies  as  much  in  the  necessity  of  the  forms  of 
thought,  as  that  of  time,  space,  quantity,  &c.,  and  perhaps  on  this 
principle  alone,  rests  the  demonstrative  certainty  of  mathematics. 

I  would  suggest  then,  whether  quantity  may  not  properly  take 
the  place  of  unity,  and  whether  its  adjuncts,  or  those  inevitable 
forms  of  the  mind  which  accompany  it,  may  not  be  wore  and  less, 
or  increase  and  decrease  of  any  quantity  which  of  itself  is  unity. 
In  such  case,  quantity, —  whether  of  force,  motion,  distance, 
space,  time,  &c.,  —  taken  at  unity,  becomes  the  standard  by  which 
to  proportion  other  quantities,  whether  they  be  greater  or  less ; 
and  this  is  to  be  done  through  the  medium  of  ratio  and  pro- 
portion as  evinced  in  the  economy  of  numbers. 

Thus  the  laws  offeree  and  motion,  as  displayed  in  time  and 
space,  are  so  adapted  to  the  econoifty  of  numbers,  exhibited  in 
their  powers  and  roots,  or  in  that  quality  or  principle  in  numbers 
which  emphatically  denotes  and  determines  by  mathematical 
precision,  ihe  ratio  or  proportion  which  exists  between  a  quan- 
tity taken  at  unity,  and  a  greater  or  less  quantity, —  (which  ratio 
or  proportion,  though  existing  in  the  mind  as  a  necessary  or 


ON    THE    LAW    OF    GRAVITY. 


109 


primitive  idea  or  thought,  is  only  definitely  determined  by  means 
of  the  powers  and  roots  of  numbers,)  —  that  unity  of  purpose  will 
always  be  discernible  in  those  laws  which  operate  in  space  and 
time,  if  unity  be  made  the  standard  by  which  to  compare  quan- 
tities; as  well  those  of  space  and  time,  as  of  the  operations  in 
space  and  time.  Thus  in  a  system  of  planets  revolving  around 
the  same  central  force,  it  is  immaterial  which  planet  we  adopt 
as  the  standard  by  assuming  its  mean  distance  at  unity,  if  we  as- 
sume its  period,  its  rate  of  velocity,  its  rate  of  motion,  &c,  each 
at  unity  ;  for  in  such  case,  the  elements  of  all  the  other  planets  of 
the  system  may  readily  be  proportioned  to  each  other,  and  to 
those  of  the  standard  planet,  or  that  whose  respective  elements 
are  called  unity.  Suppose  the  planets  A,  B  and  C,  to  revolve  in 
centric  orbits  around  the  same  centre  of  gravity  ;  A  at  the  dis- 
tance 1 ;  B  at  the  distance  2,  and  C  at  the  distance  4;  the  peri- 
od of  A  will  be  the  square  root  of  1 ;  that  of  B,  the  square  root 
of  8 ;  and  that  of  C,  the  square  root  of  64 ;  and  the  rate  of 
motion  of  A  will  be  the  square  root  of  1 ;  that  of  B  the  square  root 
of  .5;  and  that  of  C  the  square  root  of  .25;  that  is,  in  each  case, 
the  rate  of  motion  will  be  inversely  as  the  square  root  of  the 
diameter,  or  the  cube  root  of  the  inverse  of  the  period.  The  dis- 
tance of  C,  being  the  square  of  the  distance  B,  the  period  of  C 
will  be  the  square  of  the'  period  of  B  ;  the  rate  of  motion  of  G 
will  be  the  square  of  the  rate  of  motion  of  B  ;  so  also,  the  force 
of  gravity  or  attraction  applied  to  C,  will  be  the  square  of  that 
applied  to  B,  &c.  And  such  will  be  the  case  whenever  the  dis- 
tance of  one  planet  is  the  square  of  the  distance  of  another;  and 
similar  results  will  occur  if  the  distance  of  one  planet  be  the 
cube  or  fourth  power  of  the  distance  of  another  planet  of  the 
same  system. 

In  a  centric  orbit,  what  may  be  called  the  rate  of  deflection, 
is  always  inversely  as  the  square  of  the  distance,  and  is  con- 
sequently the  fourth  power  of  the  rate  of  motion ;  hence  the 
square  of  the  distance  of  the  moon  should  have  been  referred  to 
the  rate  of  deflection,  and  not  to  the  rate  of  force  which  caused 
the  deflection  ;  for  the  rates  of  deflection  of  the  planets  A  and 
C,  while  passing  over  equal  space,  will  be  inversely  as  their  dis- 
tances ;  but  while  passing  equal  time,  they  will  be  inversely  as  the 
squares  of  their  respective  distances.  Hence,  if  the  force  be  in- 
versely as  the  distance,  their  deflection,  while  passing  over  equal 
space,  will  be  as  the  force;  but  while  passing  over  equal  time, 
will  be  as  the  square  of  the  force.  But  if  the  force  be  inversely 
as  the  square  of  the  distance,  their  deflection  while  passing  over 
equal  space,  will  be  as  the  square  of  the  force ;  and  while  pass- 
ing over  equal  time,  will  be  as  the  force ;  which  in  nowise  cor- 


110  ON    THE    LAW    OF    GRAVITY. 

roborates  the  law  of  falling  bodies.  For  if  a  body  fall  from  a 
state  of  rest,  the  amount  of  fall  in  any  given  time,  is  the  square 
of  the  whole  amount  of  force  applied  from  the  commencement 
of  the  fall ;  for  the  force  applied  is  as  the  time.  Hence,  if,  in 
given  time,  the  amount  of  fall  of  one  body  from  a  state  of  rest  be 
sixteen  times  that  of  another,  the  force  that  urged  it  must  be  four 
times  as  intense  as  that  which  urged  the  other  in  its  fall ;  other- 
wise the  sum  of  all  the  spaces  passed  over  at  the  end  of  any 
given  moment,  could  not  be  the  square  of  the  sum  of  all  the 
moments  of  time  from  the  commencement  of  the  fall.  So  if  two 
bodies,  M  and  N,  fall  from  a  state  of  rest  at  the  same  time,  and 
at  the  end  of  any  given  moment,  M  shall  have  fallen  four  times 
as  far  as  N,  the  constant  force  that  urged  M  will  have  been  twice 
as  great  as  that  which  urged  N ;  but  if  M,  at  the  end  of  any 
given  moment,  shall  have  fallen  sixteen  times  as  far  as  N,  in  such 
case,  the  force  that  urged  M  will  have  been  four  times  as  great 
as  that  applied  to  N.  Let  us  now  endeavor  to  apply  these  prin- 
ciples to  the  movement  of  planets  in  their  orbits. 

In  a  centric  orbit  the  rate  of  motion  is  always  inversely  as  the 
square  root  of  the  distance :  hence,  if  the  rate  of  motion,  is 
the  square  root  of  the  force,  then  the  force  is  inversely  as  the 
distance ;  for  the  simple  reason,  that  the  inverse  of  the  distance 
is  the  square  of  the  inverse  of  the  square  root  of  the  distance. 
Hence,  in  a  centric  orbit,  because  the  rate  of  motion  is  inversely 
as  the  square  root  of  the  distance,  if  the  force  is  inversely  as  the 
square  of  the  distance,  then  the  rate  of  motion  must  be  the 
fourth  root  of  the  force  ;  — for  the  simple  reason,  that  the  inverse 
of  the  square  root  of  the  distance  is  the  fourth  root  of  the  in- 
verse of  the  square  of  the  distance.  So  in  an  eccentric  orbit,  in 
the  points  of  mean  motion  to  which  we  must  refer  the  rate  and 
progress  of  the  motion,  the  motion  is,  of  necessity,  the  inverse 
of  the  square  root  of  the  mean  distance,  as  much  as  it  is  in  a 
centric  orbit ;  else  in  a  system  of  planets,  the  squares  of  their 
periods  cannot  be  as  the  cubes  of  their  mean  distances  from  the 
sun.  Hence  in  the  points  of  mean  motion,  the  motion  will  be 
the  square  root  of  the  force,  if  the  force  is  inversely  as  the  dis- 
tance. But  if  the  force  be  inversely  as  the  square  of  the  dis- 
tance, then  in  the  points  of  mean  motion,  the  motion  will  be  the 
fourth  root  of  the  force.  And  here  it  may  be  proper  to  pause 
for  a  moment,  examine  the  original  chart,  and  get  our  latitude 
and  longitude. 

One  of  the  famous  discoveries  of  Kelper  was,  that  the  motion 
of  a  planet  revolving  in  its  orbit,  varies  in  an  inverse  ratio  as  the 
distance  varies.  This  phenomenon  only  appertains  to  eccentric 
orbits,  and  is  of  the  utmost  importance  in  the  explanation  of 


ON    THE    LAW    OF    GRAVITY.  Ill 

the  laws  of  force  and  motion,  as  applied  to  such  orbits ;  and 
had  not  its  force  and  effect,  its  spirit  and  efficacy,  been  fritterefd 
away,  and  but  the  shadow  been  retained  in  lieu  of  the  substance, 
by  those  who  succeeded  Kepler,  we  might,  long  since,  have 
been  reaping  the  benefits  which  the  discovery  of  that  principle 
seemed  so  well  calculated  to  bestow  on  science.  Yes,  frittered 
away, — until  at  length  we  find  in  place  of  it,  in  some  modern 
works  on  astonomy,  the  following,  quoted  as  Kepler's  law, 
namely,  "  The  areas  described  by  the  radius  vector  of  a  planet 
are  proportional  to  the  times."  I  am  aware  that  this  rendering 
of  the  law  is  in  accordance  with  Sir  Isaac  Newton's  notions  of 
its  importance  to  the  science  of  astronomy ;  for  Mr.  Vince  says 
that  Kepler,  after  the  discovery  of  his  said  law,  concluded,  of 
course,  that  the  radius  vector  would  describe  equal  areas  in  equal 
times,  but  was  unable  to  prove  that  it  was  so  throughout  the 
orbit ;  and  that  this  supposition  of  Kepler's  was  what  first  sug- 
gested to  Newton  the  idea  of  his  Principia ;  and  that  Newton 
proved  or  demonstrated  Kepler's  supposition  to  be  true;  and 
the  extended  consequences,  (for  good  or  for  evil,)  flowing  from 
said  demonstration,  (though  of  no  use  whatever  to  the  science 
of  astronomy,)  I  have  endeavored  in  part  to  exhibit.  Thus  the 
shadow  was  not  only  received  for  the  substance,  but  the  sub- 
stance was  perverted,  made  void,  and  useless ;  insomuch,  that 
in  lieu  of  its  being  a  variation  of  the  motion  of  a  planet  in  the 
inverse  ratio  as  the  distance  from  the  sun  varies,  it  has  been  more 
commonly  taught,  or,  at  least,  received,  that  the  motion,  or  rate  of 
motion  of  the  planet,  is  inversely  as  its  distance ;  from  which  the 
grossest  errors  have  arisen  and  been  inculcated  in  respect  to  the 
laws  of  force  and  motion.  For  there  is  not  a  simpler  or  plainer 
principle  in  respect  to  the  laws  of  motion  of  the  heavenly  bodies, 
than  that  the  rate  of  mean  motion  of  a  planet  is  inversely  as  the 
square  root  of  the  mean  distance,  and  that  when  it  revolves 
from  those  mean  points,  the  motion  varies  inversely  as  the  dis- 
tance varies.  If  it  were  not  so,  the  squares  of  the  periods  of  a 
system  of  planets  could  not  be  as  the  cubes  of  their  mean  dis- 
tances from  the  sun  ;  for  the  very  condition  of  this  law  is  based 
upon  the  principle,  that  the  mean  motion  of  a  planet  is  the  in- 
verse of  the  square  root  of  the  mean  distance ;  as  any  one  will 
discover  upon  the  slightest  examination. 

But  if  the  motion  of  a  planet  were  inversely  as  the  mean  dis- 
tance, the  squares  of  the  periods  of  a  system  of  planets  would 
be  as  the  fourth  powers  of  their  mean  distances.  And  in  such 
case,  according  to  Newton's  law  of  gravity,  namely,  "  that  force 
or  gravity  is  inversely  as  the  square  of  the  distance, "  the  force 
would  always  be  the  square  of  the  motion  in  any  part  of  the 


112  ON    THE    LAW    OF    GRAVITY.  . 

orbit ;  and  upon  this  hypothesis  has  the  base  of  astronomy  been 
laid.  '  And  hence  T  will  allege  that  the  only  principle  upon 
which  the  hypothesis  that  the  force  is  inversely  as  the  square  of 
the  distance,  could  be  sustained,  would  be,  that  the  motion  of  a 
planet  in  all  parts  of  its  orbit,  (whether  centric  or  eccentric,)  is 
inversely  as  the  distance  from  the  sun ;  in  which  case,  if  the 
force  were,  inversely  as  the  square  of  the  distance,  it  would  then, 
of  course,  be  the  square  of  the  motion.  In  such  case,  if  the  dis- 
tance were  1,  the  period  would  be  1 ;  if  the  distance  were  2, 
the  period  would  be  4,  in  lieu  of  being  2.8284  ;  if  the  distance 
were  4,  the  period  would  be  16,  or  double  what  it  really  is. 
That  is,  the  period  would  always  be  the  square  of  the  distance, 
in  lieu  of  being  the  cube  of  the  square  root  of  the  distance. 

Let  us  exemplify  the  subject  in  the  following  manner :  When- 
ever it  is  found  convenient  or  proper  to  compare  the  elements  of 
bodies  revolving  around  different  central  forces,  as  of  satellites 
about  two  primary  planets,  if  the  attractive  power  of  one  of  the 
central  forces  be  assumed  at  unity,  and  4he  other  central  force  be 
numerically  proportioned  to  unity,  we  may  then  state  the  law 
thus,  namely,  the  square  of  the  period  is  the  cube  of  the  distance 
divided  by  the  force  of  the  central  body ;  which  version  will  ap- 
ply equally  well  when  there  is  but  one  central  body  about  which 
several  bodies  revolve,  but  is  quite  convenient  when  there  are 
two  central  bodies.  Thus,  suppose  two  primary  planets  whose 
attractive  powers  are  as  1  to  4,  each  having  a  satellite  revolving 
around  it,  the  satellites  being  at  the  same  or  at  equal  distances 
from  the  centres  of  their  respective  primaries.  Let  the  primary 
whose  force  is  1  be  denoted  by  A,  and  let  the  distance,  and 
consequently  the  period  of  its  moon,  be  called  unity  or  1 ;  and 
let  the  primary  whose  force  is  4,  be  denoted  by  B.  Now  be- 
cause the  mean  force  is  the  square  of  the  mean  motion,  and  con- 
sequently is  inversely  as  the  distance ;  as  the  period  of  A's  moon 
is  1,  so  the  period  of  B's  moon  will  be  .5 ;  and  in  either  case,  it 
will  be  perceived  that  the  square  of  the  period  is  the  cube  of  the 
distance  divided  by  the  force  of  the  central  body  ;  which  law 
will  be  general,  whatever  be  the  distances  of  the  respective  satel- 
lites. 

If  it  be  true,  —  as  found  by  Euler,  LaGrange,  and  other  emi- 
nent astronomers  and  mathematicians,  so  far,  at  least,  as  could  be 
ascertained  from  observation  and  calculation,  and  even  to  that  de- 
gree of  exactness  which  induced  the  latter  to  base  certain  as- 
tronomical tables  upon  the  hypothesis,  and  of  which  he  after- 
wards remarked  that  he  saw  no  occasion  for  altering  or  changing 
his  opinion  in  respect  to  the  hypothesis,  —  that  the  power  of 
a  planet  to  attract  its  satellite  is  conjointly  as  the  bulk  or  mag- 


ON    THE    LAW    OF    GRAVITY.  113 

nitude  directly,  and  its  distance  from  the  sun  inversely;  —  then  if 
two  primary  planets  of  equal  magnitudes  revolve  about  the  sun, 
one  of  them  at  the  distance  1,  and  the  other  at  the  distance  4, 
each  having  a  satellite  revolving  around  it,  each  at  an  equal 
distance  from  the  centre  of  the  primary,  —  the  period  of  the 
satellite  of  that  planet  furthest  from  the  sun  would  be  double 
the  period  of  the  satellite  of  the  planet  nearest  the  sun  ;  and  in 
this  case,  twice  the  motion  would  require  quadruple  the  force ; 
and  in  either  case,  the  square  of  the  period  would  be  the  cube 
of  the  distance  divided  by  the  force  of  the  central  body.  Kep- 
ler's law  may  be  properly  described  thus :  The  period  is  the 
square  root  of  the  third  power  of  the  distance,  —  or  the  dis- 
tance is  the  cube  root  of  the  square  of  the  period,  —  divided 
by  the  force.  Let  us  assume  the  law  of  gravity  to  be  inversely 
as  the  distance,  and  let  three  satellites,  A,  B.  and  C,  revolve 
around  one  and  the  same  primary,  A  at  the  distance  1,  B  at  the 
distance  2,  and  C  at  the  distance  4.  Then  the  period  of  A  will 
be  1,  and  its  rate  of  motion  1 ;  the  period  of  B  will  be  2.8284, 
and  its  rate  of  motion  .707106 ;  the  period  of  C  will  be  8  and 
its  rate  of  motion  .5.  The  rate  of  force  exerted  on  A  will  be  1, 
on  B  .5,  and  on  C  .25 ;  thus  fulfilling  the  law  that  twice  the  mo- 
tion requires  four  times  the  force  to  retain  the  planet  in  its  orbit. 
Suppose  now  the  Newtonian  law  of  gravity  to  prevail  in  respect 
to  those  bodies,  namely,  that  the  force  exerted  upon  C  is  four 
times  that  exerted  upon  B,  and  sixteen  times  that  exerted  upon 
A.  In  such  case,  it  would  seem  that  inasmuch  as  B  receives 
quadruple  the  force  that  C  receives,  the  motion  of  B  should  be 
equal  to  that  of  A.  And  inasmuch  as  B  receives  but  one  fourth 
the  force  that  A  receives,  the  motion  of  B  should  be  only  equal 
to  that  of  C.  And  whether  in  such  case  B  is  placed  in  a  di- 
lemma, I  must  leave  the  world  to  judge. 

SECTION    THIRD. 

Perhaps  no  part  of  astronomy  has  caused  the  philosopher  and 
mathematician  more  perplexity  than  that  arising  from  what  is 
usually  termed  the  deflection  of  a  planet  from  a  tangent  to  its 
orbit,  accompanied  as  it  has  been  with  the  supposition  of  its 
identity,  in  its  mathematical  construction,  with  the  simple  and 
well  known  law  of  falling  bodies  near  the  earth,  and  from  the 
attempts  to  reconcile  and  harmonize  what  is  called  deflection 
from  a  tangent,  with  the  legitimate  elements  for  deducing  the 
laws  of  force  and  motion,  namely,  time,  distance,  motion  and 
gravity.  Hence  it  is  not  passing  strange  that  the  world  should 
have  received  the  doctrine  that  the  law  of  gravity  is  inversely  as 
15 


114  ON   THE   LAW    OF    GRAVITY. 

the  square  of  the  distance;  having  taken  as  their  guide  at  the 
setting  out,  a  wholly  factitious  element  in  the  laws  of  force  and 
motion,  supposing  it  to  possess  a  quality  which  it  does  not, 
namely,  identity  in  its  mathematical  or  numerical  construction 
with  the  law  of  falling  bodies  near  the  earth.  The  error,  however, 
should  not  have  been  committed,  as  I  will  endeavor  to  show ; 
even  though  deflection  is  identical  with  the  law  of  falling  bodies, 
and  for  that  purpose,  in  the  present  section,  I  will  suppose  de- 
flection to  be  identical  with  the  law  of  falling  bodies,  although 
I  think  it  demonstrable  that  it  is  not. 

For  the  purpose  of  exhibiting  Newton's  method  of  assigning 
the  law  of  gravity,  his  method  of  investigation,  and  his  re- 
sult, (which  has  generally  been  supposed  to  be  in  conformity 
with  the  law  of  falling  bodies,)  I  will  quote  from  Rees'  Cyclo- 
pedia, Vol.  XXV,  article  Moon,  wherein,  after  giving  the  magni- 
tude, motion,  density,  &c.,  compared  with  the  earth,  together  with 
the  irregularities  of  its  motions,  &c.,  the  author  proceeds  as  fol- 
lows :  "  The  irregularities  above  enumerated  and  some  others 
of  a  similar  kind,  have  been  urged  as  objections  to  the  Newton- 
ian theory  of  gravity,  though  they  were  anticipated  by  the  illus- 
trious author,,  who  not  only  evinced  their  consistency  with  it,  but 
suggested  the  explication  of  them  which  might  be  deduced  from 
that  theory  properly  understood  and  applied."  Mr.  Rees,  after 
quoting  from  Mr.  Vince's  astronomy  on  the  same  subject,  pro- 
ceeds :  "  Mr*  Euler  also  retracted  his  own  erroneous  opinion  in 
deference  to  the  judgment  of  Mr.  Clairaut,  and  concurs  with 
him  in  doing  ample  justice  to  the  Newtonian  theory,"  and  ob- 
serves that  "  others,  and  particularly  Mr.  Machin  and  Mr.  Frisi, 
have  prosecuted  a  similar  investigation  of  this  theory  and  con- 
tributed to  establish  it."  Mr.  Rees  then  proceeds  in  this  very 
remarkable  manner :  "  It  would  not  be  consistent  with  the  limits 
nor  the  nature  of  this  work,  to  investigate,  by  tedious  and  elabor- 
ate processes,  of  an  analytical  and  geometrical  kind,  the  various 
equations  that  have  been  explored  for  the  illustration  of  these 
laws.  Much  has  been  done  in  this  way  by  several  learned  math- 
ematicians, and  of  late  by  Professor  Vince,  who  is  eminently 
qualified  for  the  undertaking;  and  we  shall  therefore  refer  the 
reader  who  may  be  desirous  of  further  information,  and  who  has 
no  access  to  a  variety  of  other  publications,  to  the  second  volume 
of  Vince's  Complete  System  of  Astronomy,  Chapter  XXXII." 
Mr.  Rees  proceeds  :  "  We  shall,  however,  in  this  place,  introduce 
a  general  view  of  the  Newtonian  theory  of  gravity,  as  it  is  ap- 
plied to  the  solution  of  the  investigations  of  the  moon's  motions." 
"  We  have  already,  under  the  article  Gravitation,  illustrated  and 
confirmed  the  Newtonian  theory  of  gravity,  as  it  regards  the 


ON    THE    LAW    OF    GRAVITY.  115 

moon  and  other  planets ;  but  as  the  subject  is  of  importance,  and 
as  it  is  immediately  connected  with  what  follows,  we  shall  here 
give  a  concise  statement  of  the  leading  facts  by  which  the  iden- 
tity of  the  centripetal  force  as  it  respects  the  moon,  and  that  of 
gravity,  was  originally  explained  and  established,  referring  for  a 
more  detailed  account,  to  the  article  just  cited."  "  It  is  well 
known  and  universally  allowed  that  the  planets  are  retained  in 
their  orbits  by  some  power  which  is  constantly  acting  upon  them ; 
that  this  power  is  directed  towards  the  centres  of  their  orbits ; 
that  the  efficacy  of  this  power  increases  upon  an  approach  to 
the  centre,  and  diminishes  by  its  recess  from  the  same,  and  that 
it  increases  according  to  a  certain  law,  namely,  that  of  the 
squares  of  the  distances,  as  the  distance  diminishes ;  and  that 
diminishes  in  the  same  manner  as  the  distance  increases.  Now 
by  comparing  this  centripetal  force  of  the  planets  with  the  force 
of  gravity  on  the  earth,  they  will  be  found  perfectly  alike.  This 
we  shall  illustrate  in  case  of  the  moon,  the  nearest  to  us  of  all 
the  planets." 

u  The  rectilinear  spaces  described  in  any  given  time  by  a  falling 
body,  urged  by  any  powers,  reckoning  from  the  beginning  of  its 
descent,  are  proportionable  to  those  powers.  Consequently  the 
centripetal  force  of  the  moon,  revolving  in  its  orbit,  will  be  to 
the  force  of  gravity  on  the  surface  of  the  earth,  as  the  space  which 
the  moon  would  describe  in  falling  any  little  time,  by  her  cen- 
tripetal force  towards  the  earth,  if  she  had  no  circular  motion  at 
all,  to  the  space  which  a  body  near  the  earth  would  describe  in 
falling  by  its  gravity  towards  the  same."  "  By  a  very  easy  and 
obvious  calculation  of  these  two  spaces  it  will  appear  that  the 
first  of  them  is  to  the  second,  —  i.  e.,  the  centripetal  force  of  the 
moon  revolving  in  her  orbit  is  to  the  force  of  gravity  on  the  sur- 
face of  the  earth,  —  as  the  square  of  the  earth's  semi-diameter  to 
the  square  of  the  semi-diameter  of  her  orbit,  which  is  the  same 
ratio  as  that  of  the  moon's  centripetal  force  in  her  orbit  to  the 
same  force  near  the  surface  of  the  earth.  The  moon's  centripetal 
force  is  therefore  equal  to  the  force  of  gravity.  These  forces 
consequently  are  not  different,  but  they  are  one  and  the  same  ; 
for  if  they  were  different,  bodies  acted  upon  by  the  two  pow- 
ers conjointly,  would  fall  towards  the  earth  with  a  velocity 
double  to  that  arising  from  the  sole  power  of  gravity."  "  It  is 
evident,  therefore,  that  the  moon's  centripetal  force,  by  which  she 
is  retained  in  her  orbit,  and  prevented  from  running  off  in  tan- 
gents, is  the  power  of  gravity  of  the  earth  extended  thither." 

Mr.  Rees,  then,  after  the  manner  of  others,  supposes  that  the 
moon  is  drawn  from  its  tangent  towards  the  centre  of  the  earth 
about  15f  3-  Paris  feet  in  a  minute  of  time,  which  is  the  same 


116  ON    THE    LAW    OF    GRAVITY. 

distance  a  body  near  the  surface  of  the  earth  will  fall  in  one  sec- 
ond of  time,  and  proceeds  thus :  "  But  this  force  (as  applied  to 
the  moon)  being  known  from  the  elliptical  figure  of  her  orbit  to 
be  reciprocally  proportional  to  the  square  of  the  distance,  would 
impel  the  moon,  supposed  to  be  at  the  surface  of  the  earth, 
through  a  space  equal  to  60  X  60  X  15^  feet  in  one  minute." 
"  But  bodies  impelled  by  the  force  of  gravity  fall  near  the  sur- 
face of  the  earth  through  the  space  of  15^  Paris  feet  in  one 
second,  and  the  spaces  being  as  the  squares  of  the  time,  through 
60  X  60  X  15TV  in  a  minute.  Consequently,  as  the  force  by 
which  the  moon  is  retained  in  its  orbit,  and  the  force  of  gravity, 
produce  the  same  effects  in  the  same  circumstances,  they  are 
the  same  forces." 

This  then  is  the  law  of  gravity,  as  promulgated  by  Sir  Isaac 
Newton,  by  which  the  heavenly  bodies  move  in  their  eternal 
rounds.  I  will  therefore  give  a  short  version  of  the  foregoing 
method  to  suit  my  own  purposes  and  my  own  method  of  inves- 
tigation, and  will  in  due  time,  enter  upon  a  confutation  of  his 
deductions,  and  upon  the  support  of  my  own.  Thus  a  body 
commencing  its  fall  near  the  surface  of  the  earth  will,  in  the  first 
second  of  time,  (which  may  be  called  the  first  moment,)  fall 
about  15T^  Paris  feet,  which  may  properly  be  called  one  space ; 
in  the  second  moment  the  body  will  fall  over  three  like  spaces ; 
over  five  like  spaces  in  the  third  moment,  and  so  on;  the 
spaces  increasing  with  the  odd  numbers  in  their  order.  Hence 
if  we  set  down  all  the  common  numbers  in  their  order,  as  1,  2,  3, 
4,  and  so  on,  ad  infinitum,  and  extract  the  square  root  of  such 
series,  taking  figure  1  at  the  left  hand  as  the  whole  of  the  first 
period  in  the  process  of  the  extraction,  then  the  square  root  of 
such  series  will  be  expressed  by  placing  figure  1  over  each  odd 
number  in  the  series,  in  which  case  we  shall  have  a  very  con- 
venient formula  whereby  to  express  the  law  of  falling  bodies 
near  the  earth ;  in  which,  the  square  root  of  the  series  constantly 
expressed  by  figure  1,  will  denote  the  individual  and  equal  mo- 
ments, (and  also  the  equal  forces  in  those  moments,)  in  which  the 
body  shall  have  been  falling  at  the  end  of  any  given  moment. 
The  sum  of  all  the  odd  numbers  of  the  series  will  be  the  whole 
amount  of  spaces  over  which  the  body  has  fallen  at  the  end  of  any 
given  moment,  which  will  be  the  square  of  the  sum  of  all  the  mo- 
ments or  of  the  forces  in  those  moments  in  which  the  body  has 
been  falling ;  and  the  even  numbers  in  the  series  will  denote  the 
acquired  motion,  or  the  rate  of  motion  at  the  end  of  any  given 
moment.  The  moon  revolves  about  the  earth  in  an  orbit  60 
times  as  far  from  the  centre  of  the  earth  (or  centre  of  gravity)  as 
the  surface  of  the  earth  is ;  and  we  will  suppose  it  to  be  found 


ON    THE    LAW    OF    GRAVITY.  117 

from  observation  and  calculation,  that  the  fall  of  the  moon  from 
a  tangent  to  its  orbit,  towards  the  centre  of  the  earth,  is  one 
space,  or  15T^  feet,  in  60  minutes,  and  hence  that  the  fall  of  the 
moon,  at  her  distance  from  the  earth,  is  but  3^0  part  as  rapid  as 
that  of  a  body  at  the  surface  of  the  earth. 

And  from  premises  like  these  the  deduction  has  been  drawn 
by  Newton  and  endorsed  by  others,  that  the  force  is  inversely  as 
the  square  of  the  distance ;  notwithstanding  the  supposed  iden- 
tity shown  between  the  centripetal  force  which  retains  the  moon 
in  its  orbit,  and  the  law  of  gravitation  at  the  surface  of  the  earth 
which  causes  bodies  to  fall  and  to  accelerate  their  motion  during 
the  fall.  And  hence  it  has  been  concluded  that  the  force  of  the 
earth's  attraction  at  its  surface  is  3600  times  as  great  as  at  the 
distance  of  the  moon,  when  in  fact  it  is  but  60  times  as  great ; 
as  I  propose  to  show  by  a  very  simple  process.  And  had 
Clairaut  and  Euler  but  taken  a  common  sense  view  of  the  sub- 
ject, (having  the  elements  all  before  them,)  it  would  not  have 
been  necessary  for  them  to  have  retracted  their  rational  doubts 
as  to  the  Newtonian  law  of  gravity,  or  to  have  resorted  to  what 
is  called  the  problem  of  the  three  bodies,  or  any  other  mysterious 
problem,  for  the  purpose  of  accounting  for  the  motion  of  the 
moon's  apogee,  in  corroboration  of  a  law  as  simple  as  it  is 
universal. 

In  the  present  investigation  of  the  subject,  I  will  make  use 
of  the  element  of  time,  in  the  abstract,  as  that  of  equal  moments, 
in  lieu  of  our  arbitrary  division,  as  that  of  seconds,  minutes,  &c., 
each  of  which  moments  being  assumed  at  unity,  or  as  a  unit  of 
time,  may  be  rationally  and  readily  compared  with  the  other  re- 
quisite elements  in  the  investigation,  each  of  which  will  be  used 
in  the  abstract,  and  each  assumed  at  unity ;  and  such  method 
is  as  applicable  to  the  law  of  falling  bodies  as  to  an  investi- 
gation of  planetary  motion.  Now,  "  the  squares  of  the  periods 
of  the  planets  are  as  the  cubes  of  their  mean  distances  from  the 
sun" — is  emphatically  the  great  astronomical  chronometer  from 
which  the  most  simple  and  rational  deduction  can  be  drawn  in 
respect  to  the  law  of  gravity,  and  its  application  to  planetary  mo- 
tion. Thus  if  unity  be  made,  (as  it  emphatically  is,)  the  base 
or  centre  of  all  ratio,  then,  in  respect  to  any  one  planet  of  a 
system,  if  one  of  the  elements  used  in  any  investigation  be  as- 
sumed at  unity,  each  and  every  other  of  the  elements  must  be 
assumed  at  unity,  else  the  square  of  the  period  will  not  be  the 
cube  of  the  distance. 

Now,  our  great  object  of  inquiry  is,  what  is  the  rate  or  ratio  of 
the  force  of  gravity  to  that  of  distance  from  the  centre  of  gravity  ? 
This  then  naturally  suggests,  as  the  first  or  standard  element,  that 


118  ON    THE    LAW    OF    GRAVITY. 

of  passive  distance  in  space,  which  has  nothing  whatever  to  do 
in  the  performance  of  the  operation.  Nevertheless,  a  ponderous 
body  is  not  passed  over,  or  made  to  describe  a  given  quantity  of 
space,  except  in  some  given  quantity  of  passive  time ;  hence 
passive  time  may  properly  be  regarded  as  the  second  element  to 
be  considered. 

The  third  element  is  that  of  the  given  space  to  be  passed  over 
or  described  in  a  given  time  at  a  given  distance,  and  is  conse- 
quently a  passive  element. 

The  fourth  element  is  that  of  motion,  and  is  of  itself  an  effect, 
an  operation,  being  the  phenomenon  observed ;  the  rate  of  which 
is  to  be  proportioned  to  the  rate  of  motion  at  a  different  dis- 
tance. 

The  fifth  element  is  that  of  the  force  of  gravity  which  causes 
the  phenomenon,  or  passage  of  the  body  over  the  given  space, 
or  which  controls  its  passage  over  the  given  space,  the  rate  of 
which  is  to  be  proportioned  to  the  rate  of  force  at  a  different 
distance,  and  is  to  be  deduced  from  a  proper  consideration  of 
the  other  elements,  and  more  especially  from  a  proper  con- 
sideration of  the  different  rates  of  motion,  (or  of  the  observed 
phenomena,)  at  different  distances  from  the  centre  of  gravity. 
And  whether  we  attempt  to  deduce  the  rate  of  force  or  gravity 
as  proportioned  to  distance,  from  a  consideration  of  two  or  more 
planets  revolving  in  their  orbits  at  different  distances  from  the 
sun ;  or  from  a  consideration  of  two  or  more  bodies  commenc- 
ing their  fall  towards  the  earth  at  the  same  time,  but  from  differ- 
ent distances  from  the  centre  of  the  earth ;  that  planet,  or  that 
body  whose  distance  is  assumed  at  unity,  must  necessarily  have 
each  and  every  one  of  its  elements  assumed  at  unity,  by  which 
the  elements  of  the  other  planet  or  body  may  be  compared  and 
proportioned,  as  with  ratio  in  unity.  Notwithstanding,  if  two 
planets  revolving  round  the  sun  at  different  distances,  or  if  two 
bodies  commencing  their  fall  at  the  same  time  from  differ- 
ent distances  from  the  centre  of  the  earth,  in  which  each  and 
every  element  must  be  assumed  at  unity  for  that  planet  or  fall- 
ing body  whose  distance  is  assumed  at  unity ;  nevertheless,  as 
what  is  termed  the  amount  of  deflection  of  a  planet  from  a  tan- 
gent to  the  orbit  (even  in  any  little  time)  while  revolving  in  the 
orbit,  is  not  the  same,  or  not  equal  in  amount  to  what  the  recti- 
linear fall  of  a  body  from  a  state  of  rest  towards  the  centre  of 
gravity  from  the  point  of  contact  of  the  tangent  and  orbit,  would 
be  in  the  same  given  time,  however  small,  (although  they  were 
assumed  to  be  equal  by  Sir  Isaac  Newton,)  such  difference 
will  cause  a  much  greater  dissimilarity  between  the  methods  of 


ON    THE    LAW    OF    GRAVITY.  119 

deducing  the  law  of  gravity  from  a  consideration  of  two  or 
more  planets,  and  the  law  of  falling  bodies,  than  would 
otherwise  exist.  But  notwithstanding  the  amount  of  deflection 
of  a  planet  from  a  tangent,  even  in  any  little  time,  is  equal  to 
the  versed  sine  of  the  arc  described  by  the  planet  in  such  little 
time,  yet  such  versed  sine,  being  greater  in  amount  or  not  of 
the  same  amount  of  space,  as  would  be  described  by  a  body 
falling  from  a  state  of  rest  towards  the  centre  of  gravity  from 
that  distance  in  the  same  little  time,  would  seem  very  strongly 
to  suggest  that  a  like  discrepancy  must  exist  between  any  nu- 
merical development  of  the  law  of  falling  bodies,  and  that  which 
is  generally  supposed  to  give  a  rational  approximation  of  the 
ratio  of  the  circumference  of  the  circle  to  the  diameter,  as  would 
exist  between  that  of  falling  bodies  and  the  deflection  of  a 
planet  from  a  tangent  to  its  orbit.  Nevertheless,  either  of  these 
methods  should  have  led  to  the  conclusion  that  the  force  of 
gravity  varies  inversely  as  the  distance  varies ;  and  it  may  not 
be  amiss  here  to  give  some  few  hints  as  to  the  discrepancy 
above  named,  that  the  mind  may  readily  make  the  application 
in  respect  to  such  discrepancy. 

Conceive  then  the  planets  A  and  C  revolving  in  centric  orbits 
about  the  sun ;  A  at  the  distance  1,  and  C  at  the  distance  4 ;  and 
conceive  a  straight  line  to  pass  from  the  centre  of  the  sun,  inter- 
secting both  orbits,  called  the  intersecting  line  ;  let  a  tangent  to 
each  orbit  pass  from  the  intersecting  line  in  the  direction  in 
which  the  planets  are  to  revolve,  —  and  let  the  two  planets  be 
conceived  to  start  from  the  intersecting  line  at  the  same  instant 
of  time. 

Then  if  A  shall  have  passed  ever  so  small  a  distance  in  its  or- 
bit, from  the  intersecting  line,  C  will  have  passed  one  half  the  dis- 
tance that  A  has.  Now  some  small  distance  from  the  intersect- 
ing line  on  the  orbit  of  A,  may  be  called  a  unit  of  space  ;  and 
the  time  in  which  A  is  describing  it  may  be  called  a  unit  of 
time ;  hence  in  one  unit  or  moment  of  time,  C  will  describe  half 
a  unit  of  space,  or  in  two  units  or  moments  of  time,  will  describe 
one  unit  of  space  ;  and  when  A  has  passed  over  one  moment  or 
one  unit  of  space  from  the  intersecting  line,  the  amount  of  its  de- 
flection from  the  tangent  will  be  called  unity  or  1.  And  hence, 
if  such  deflection  be  governed  by  the  law  of  falling  bodies,  when 
A  has  passed  over  two  units  of  space,  its  amount  of  deflection 
will  then  be  4,  or  four  times  that  at  the  end  of  one  space.  But 
as  the  orbit  of  C  is  four  times  as  dilate  as  that  of  the  orbit  of  A, 
if  the  law  of  falling  bodies  were  to  prevail  according  to  the 
hypothesis  of  Sir  Isaac  Newton,  then,  when  C  has  passed  one 
unit  of  space,  its  amount  of  deflection  would  be  just  one  fourth 


120  ON    THE    LAW    OF    GRAVITY. 

of  that  of  A  at  the  end  of  one  space,  and  when  C  has  passed 
over  one  moment  or  unit  of  time,  its  amount  of  deflection  would 
be  just  one  sixteenth  that  of  A  when  A  has  passed  over  one 
unit  of  time.  If  such  be  the  case,  (and  it  is  so,  excepting  the 
variation,)  then  at  the  end  of  equal  time,  the  force  applied  to  C 
is  (in  the  whole  amount)  one  fourth  of  that  applied  to  A  in  the 
same  time,  for  the  reason  that  it  would  produce  one  sixteenth  of 
the  deflection  in  C  that  the  force  applied  to  A  gives  in  the  same 
time.  And  when  C  and  A  have  passed  over  equal  spaces,  the 
deflection  of  C  will  then  be  one  fourth  that  of  A,  which,  if 
strictly  true,  would  perfectly  agree  with  the  law  of  falling  bodies. 
It  is  manifest,  then,  that  there  is  a  resemblance  between  deflec- 
tion and  the  law  of  falling  bodies ;  and  were  they  identical,  then 
the  law  of  deflection  would  be  sufficient  to  prove  conclusively 
that  the  rate  of  force  or  gravity  applied  to  C,  is  one  fourth  of 
that  applied  to  A,  which  rate  of  force  properly  refers  to  time, 
viz.,  a  continuation  of  the  force  long  enough  to  make  up  a  given 
amount.  Deflection,  however,  does  so  far  resemble  the  law  of 
falling  bodies,  that  those  who  undertook  to  prove  them  identical, 
and  to  deduce  the  law  of  gravity  therefrom,  should  have  referred 
the  square  of  the  distance  to  the  amount  of  deflection  in  time,  in 
lieu  of  referring  it  to  the  rate  or  intensity  of  the  force  of  gravity  ; 
for  the  rate  of  deflection  in  time  would  then  have  been  assumed 
as  the  inverse  of  the  square  of  the  distance,  and  as  the  square 
of  the  rate  of  gravity  directly,  as  in  the  case  of  falling  bodies. 
If  then  deflection  were  identical  with  that  of  falling  bodies,  then, 
in  either  case,  the  amount  of  deflection  or  fall  from  the  beginning 
would  always  be  the  square  of  the  whole  amount  of  force  ex- 
pended during  such  deflection  or  fall. 

In  such  case,  the  amount  of  deflection  being  as  the  square  of 
the  force  directly,  and  inversely  as  the  square  of  the  distance, 
the  rate  of  force  must  necessarily  be  inversely  as  the  distance,  as 
the  same  is  determined  by  the  law  of  falling  bodies.  Here  then 
rested  the  Newtonian  error,  in  referring  the  square  of  the  dis- 
tance to  the  force,  instead  of  referring  it  to  the  fall  or  deflection 
in  time,  to  which  it  manifestly  relates.  Thus,  to  refer  again  to 
the  planets  A  and  C,  and  suppose  deflection  and  the  law  of  fal- 
ling bodies  to  be  identical,  A  would  receive  twice  the  amount 
of  force  that  C  would,  while  passing  over  equal  space,  because 
the  deflection  of  A  would  then  be  quadruple  to  that  of  C ;  and 
A  would  receive  quadruple  the  amount  of  force  that  C  would 
while  passing  over  equal  time,  because  the  deflection  of  A  would 
then  be  sixteen  fold  that  of  C.  Hence  A  receives  double  the 
force  in  half  the  time  ;  or  C  receives  half  the  force  in  double  the 
time  ;  the  consequence  of  which  is,  that  in  equal  time  A  receives 


ON    THE    LAW    OF    GRAVITY.  121 

four  times  the  amount  of  force  that  C  does ;  or  C  receives 
one  fourth  of  the  amount  of  force  that  A  does  in  equal  time ; 
hence  if  the  time  of  C  be  4,  and  the  time  of  A  1,  they  will  then 
each  have  received  a  like  amount  of  force  ;  proving  that  the  rate 
of  force  is  inversely  as  the  distance  ;  that  the  rate  of  force  ap- 
plied to  A  is  fourfold  that  applied  to  C ;  and  as  the  time  of  the 
period  of  C  is  eight  times  that  of  A,  hence  C  will  receive,  in  one 
of  its  entire  revolutions  or  in  the  time  of  its  period,  double  the 
amount  of  force  that  A  will  receive  in  the  time  of  A's  period. 
But  if  the  rate  of  force  were  inversely  as  the  square  of  the  dis- 
tance, then  C,  in  the  time  of  its  period,  would  receive  just  half 
the  amount  of  force  that  A  would  receive  in  the  time  of  A's 
period.  If  the  amount  of  deflection  of  C  were  just  four  times 
as  great  in  two  units  of  time  as  it  is  in  one,  (in  accordance  with 
the  law  of  falling  bodies,)  then  the  amount  of  C's  deflection  in 
one  unit  of  time  would  be  the  square  of  the  amount  of  C's  de- 
flection in  one  unit  of  space.  The  distance  of  C  being  4,  its  pe- 
riod is  consequently  8,  otherwise  it  would  not  fulfil  the  Keplerean 
law ;  hence  if  the  distance  of  another  planet  be  the  square  root 
of  the  distance  of  C,  the  period  of  such  planet  will  necessarily 
be  the  square  root  of  the  rate  of  motion  of  C;  and  the  same  is 
true  of  its  rate  of  force  or  gravity ;  and  also  of  its  deflection  in 
time.  And  similar  will  be  the  case  whenever  the  distance  of 
one  planet  shall  be  the  square  root  of  the  distance  of  another 
planet  of  the  same  system ;  for  such  result  only  will  fulfil  the 
great  law  of  Kepler. 

Hence  the  distance  of  a  planet  divided  by  the  rate  of  its  mean 
motion  gives  the  time  of  its  period  ;  or  the  distance  divided  by 
the  time  of  the  period,  gives  the  rate  of  mean  motion.  Hence 
the  mean  distance  is  always  the  square  of  the  reciprocal  of  the 
mean  motion ;  or  the  mean  motion  is  the  reciprocal  of  the  square 
root  of  the  mean  distance,  or  of  the  cube  root  of  the  time  of  the 
period.  We  find  then  these  proportions  to  prevail  between  the 
rate  of  motion  and  the  time  of  the  period,  viz.,  when  the  dis- 
tance is  1,  the  rate  of  motion  is  equal  to  the  period  ;  when  the 
distance  is  2,  the  rate  of  motion  is  equal  to  l-4th  of  the  period  ; 
when  the  distance  is  3,  the  rate  of  motion  is  equal  to  l-9th  of  the 
period,  &c. ;  or  in  other  words,  the  distance  divided  by  the  time 
of  the  period  gives  the  rate  of  motion  ;  and  the  rate  of  motion 
is  always,  and  by  every  consideration  which  may  be  given  to 
the  subject,  inversely  as  the  square  root  of  the  distance  ;  for  this 
it  is,  which  establishes  the  phenomena  according  to  Kepler's 
law.  Hence  if  we  take  three  planets,  A,  B  and  C, —  A  at  the 
distance  1,  B  at  the  distance  2,  C  at  the  distance  4,  —  the  time 
required  for  B  to  pass  over  a  space  equal  to  the  circumference  of 
16 


122  ON    THE    LAW    OP    GRAVITY. 

A's  orbit,  will  be  equal  to  the  square  root  of  B's  distance,  viz., 
the  square  roo*  of  2 ;  and  the  time  required  for  C  to  pass  over  a 
like  space  will  oe  equal  to  the  square  root  of  C's  distance,  viz., 
the  square  root  of  4;  the  motion,  or  rate  of  motion  of  each 
planet,  being  the  reciprocal  of  the  square  root  of  its  own  distance. 
And  because  the  rate  of  motion  is  inversely  as  the  square  root 
of  the  distance,  hence  if  the  rate  of  the  force  of  gravity  be  in- 
versely as  the  square  of  the  distance,  the  rate  of  force  or  gravity 
will  be  the  fourth  power  of  the  rate  of  motion.  But  if  the  rate 
of  force  or  gravity  be  inversely  as  the  distance,  then  the  rate  of 
force  or  gravity  will  be  the  square  of  the  rate  of  motion,  in  ac- 
cordance with  all  we  know  in  respect  to  the  laws  of  force  and 
motion  as  resulting  from  a  consideration  of  mechanics  or  dynam- 
ics, as  also  with  well  established  facts  which  have  long  been  as- 
sented to  by  mathematicians  and  astronomers,  in  matters  wholly 
analogous  and  corroborative :  as  that  twice  the  motion  requires 
quadruple  the  force,  whether  it  be  the  result  of  an  impulsive  force 
or  of  that  constant  and  uniform  force  which  retains  a  planet  in  a 
centric  orbit ;  or  produced  in  any  other  manner  in  which  mo- 
tion may  be  conceived  to  be  the  result  of  force.  In  respect  to 
the  difference  in  point  of  identity  between  deflection  from  a  tan- 
gent, and  the  law  of  falling  bodies,  inasmuch  as  I  shall  have  oc- 
casion hereafter  to  refer  to  some  direct  and  conclusive  evidence 
on  this  point,  I  will  defer  the  same  for  the  present  and  proceed  to 
a  short  consideration  of  the  law  of  falling  bodies,  from  which 
the  law  of  gravity  may  also  be  deduced. 

In  a  consideration  of  the  law  of  falling  bodies,  the  force  ap- 
plied is  to  be  considered  as  constant  and  as  uniform  as  the  time 
in  which  the  force  is  applied ;  and  the  whole  amount  of  force 
applied  while  the  body  is  describing  a  given  space,  may  be  di- 
vided into  as  many  equal  parts  as  the  whole  amount  of  time  is 
divided  into  in  which  the  body  is  describing  the  given  space. 

The  time  required  for  a  body  to  fall  from  a  state  of  rest  over  a 
given  space  (one  unit  of  space)  is  as  the  distance,  by  all  determi- 
nations of  the  law  of  falling  bodies. 

Hence  the  rate  or  intensity  of  the  force  bears  the  same  ratio  or 
proportion  to  the  distance  that  it  does  to  the  time.  It  is  as  much 
less  or  more  than  the  distance  as  it  is  less  or  more  than  the  time. 
If  the  rate  or  intensity  of  the  force  which  urges  a  body  over  a 
given  space,  or  unit  of  space,  be  inversely  as  the  distance,  it  is 
also  inversely  as  the  time.  If  it  be  inversely  as  the  square  of  the 
distance,  it  will  also  be  inversely  as  the  square  of  the  whole 
time,  —  thus  fulfilling  the  whole  law  in  respect  to  the  mechanical 
powers. 

If  A  and  B  are  to  fall  from  a  state  of  rest  at  the  same  time,  A 
from  the  distance  1  from  the  centre  of  gravity,  and  B  from  the 


ON    THE    LAW    OF    GRAVITY.  123 

distance  60,  (each  body  to  describe  a  like  amount  of  space, 
namely,  a  unit  of  space,)  B,  or  the  body  at  the  distance  60,  will 
require  sixty-fold  as  long  time  as  A,  or  the  body  at  the  distance 
1.  Hence  if  the  labor  to  be  performed,  (namely,  the  unit  of  space 
to  be  described,)  be  at  sixty-fold  the  distance  from  the  centre  of 
gravity,  (or  centre  of  force,)  the  force  at  that  distance  is  propor- 
tionally so  much  less  than  at  the  distance  1,  that  sixty-fold  the 
time  is  required  that  would  be  required  at  the  distance  1,  for  per- 
forming the  same  amount  of  labor. 

If,  then,  we  proportion  the  rate  or  intensity  of  the  force  to  the 
whole  amount  of  the  time  passed  over,  we  thereby  proportion  the 
rate  or  intensity  of  the  force  to  the  whole  amount  of  the  distance 
from  the  centre  of  gravity.  The  space  then  to  be  described,  or 
the  labor  to  be  performed,  being  unity,  the  length  of  time  deter- 
mines the  distance,  or  the  distance  determines  the  length  of  time. 
And  until  we  are  able  to  obtain  some  kind  of  evidence  that  the 
great,  eternal,  and  universal  law, — that  the  intensity  of  the  force  is 
diminished  as'the  whole  time  is  increased,  qr  that  the  rate  or  in- 
tensity of  the  force  must  be  increased  as  the  whole  time  shall  be 
diminished,  in  order  that  the  task  or  labor  shall  be  performed, — 
has  been  altered,  we  had  better  abide  by  the  law  as  it  was  in  the 
beginning,  is  now,  and  ever  shall  be.  The  following  numerical 
formula  is  in  accordance  with  those  generally  given  for  denoting 
or  explaining  the  law  of  falling  bodies  towards  the  earth,  —  and 
the  same  may  also  serve  to  indicate  how  extremely  simple  those 
demonstrations  are,  and  always  have  been,  which  lie  at  the  foun- 
dation of  true  science,  and  have  rendered  the  greatest  benefits ; 
and  hence,  important  demonstrations  are  not  enveloped  in  a 
cloud  of  darkness  beyond  the  ken  or  sagacity  of  most  mortals  to 
penetrate  ;  and  whenever  what  may  be  alleged  to  have  been  de- 
monstrated is  so  clothed,  it  is  to  be  feared  that  it  is  but  the  at- 
tempt to  equate  error  to  correspond  with  truth,  which  must  ne- 
cessarily require  a  mysterious  process.  It  will  be  seen,  also,  that 
the  following  formula  is  only  intended  to  denote  the  numerical 
law  of  a  falling  body  in  reference  to  the  manner  in  which  time  is 
apportioned  to  any  given  amount,  of  space  described  or  passed 
over  by  the  body  in  its  fall  from  a  state  of  rest,  without  regard  to 
distance  from  a  centre  of  gravity,  and  consequently  without  regard 
to  any  rate,  intensity  or  efficiency  of  force  by  which  the  body 
should  be  urged,  and  without  intention  to  ascertain  the  rate  or 
intensity  of  gravity  as  proportioned  to  distance  from  a  centre  of 
gravity ;  for,  short  of  a  comparison  of  certain  elements  of  two 
falling  bodies,  the  rate  of  gravity  could  not  be  proportioned  to 
the  distance. 

In  accordance  with  the  popular  formula  above  named,  the 
upper  series,  composed  of  number  1,  repeated,  may  be  called  the 


124  ON    THE    LAW    OF    GRAVITY. 

uniform  series,  and  is  used  to  divide  the  whole  amount  of  time 
in  which  a  body  is  falling  over  a  given  space,  into  equal  parts  or 
moments,  as  seconds,  for  instance. 

The  second  column,  composed  of  the  odd  numbers  in  their 
order,  may  be  called  the  accelerating  series,  —  the  sum  of  which 
odd  numbers  at  the  end  of  any  given  moment  of  the  time,  will 
always  be  the  square  of  the  number  of  moments  of  time  in  which 
the  body  has  been  falling ;  or,  in  other  words,  the  amount  of 
space  described  or  passed  over  at  any  given  instant  of  time  from 
the  commencement  of  the  fall,  is  the  square  of  the  whole  time 
that  has  been  passed  over  from  the  commencement  of  ihe  fall. 
Thus  the  whole  space  passed  over  during  the  amount  of  time 
allowed  for  the  fall,  is,  by  the  series  of  odd  numbers,  divided  into 
as  many  equal  parts  (called  spaces)  as  will  make  the  square  of 
the  number  of  equal  moments  into  which  the  whole  time  is 
divided ;  the  first  space,  or  that  denoted  by  unity,  or  figure  1,  be- 
ing the  amount  of  space  which  the  body  will  describe  in  the  first 
moment  of  time ;  and  the  law,  numerically  expressed,  is,  that  the 
body  will  describe  three  times  the  space  in  the  second  moment 
that  it  does  in  the  first,  —  five  times  in  the  third  moment,  and 
so  on. 

Another  part  of  the  law,  in  respect  to  a  single  falling  body,  is 
numerically  expressed  by  the  even  numbers  in  their  order,  de- 
noting the  rate  of  velocity  or  motion  acquired  at  the  end  of  any 
given  moment  of  the  fall.  The  acquired  motion  at  the  end  of 
any  given  moment  of  time,  being  sufficient,  without  further  force 
applied,  to  carry  the  body  over  one  more  space  in  the  n  xt  suc- 
ceeding moment  than  it  passed  over  in  such  given  moment. 
But  by  the  aid  of  the  constant  force  applied,  the  body  will  pass 
over  two  more  additional  spaces  in  each  successive  moment, 
thus  increasing  the  number  of  spaces  as  the  odd  numbers  in- 
crease ;  and  hence  we  find  that  the  acquired  motion  is  as  ac- 
tive as  the  force  is  efficient. 

A  formula,  however,  designed  to  express  numerically  the  law 
of  a  falling  body,  may  always  be  most  properly  expressed  in  two 
series  or  lines  of  figures,  namely,  the  uniform  series,  or  that  ex- 
pressed by  the  constant  figure  1,  and  the  series  of  natural  num- 
bers ;  figure  1  occurring  only  once  in  each  odd  number  of  the 
series  of  natural  numbers.  Then  the  uniform  series  to  any  extent 
will  be  the  square  root  of  the  series  of  natural  numbers  to  the  same 
extent,  in  case  we  make  figure  1,  in  the  natural  numbers,  the  first 
period  in  the  process  of  extraction  ;  and  in  such  case  the  even  num- 
bers in  the  series  of  natural  numbers,  (namely,  those  figures  which 
denote  the  rate  of  acceleration,)  will  always  come  out  as  a  remain- 
der in  the  process  of  extracting  the  square  root,  or  uniform  series 


ON    THE    LAW    OF    GRAVITY.  125 

The  formula,  with  the  uniform  series  carried  or  extended  to 
eight  places,  may  be  thus  : 


11111  1  1  1 

1    2    3    4    5    6    7    8    9    10   11  12   13  14    15    16 


Now  it  is  manifest  that  in  calculating  the  law  of  the  fall 
simply  of  an  individual  body  from  a  state  of  rest,  —  in  which 
case  distance  from  the  centre  of  the  earth  or  centre  of  gravity  is 
not  material,  for  the  reason  that  the  law,  numerically  expressed, 
will  be  the  same,  whether  the  force  be  more  or  less,  (so  that  it 
be  uniform,)  —  we  may  always  consider  the  distance  from  the 
centre  of  gravity  to  be  unity,  whether  it  be  at  the  surface  of  the 
earth  or  at  the  distance  of  the  moon  ;  and  that  it  is  only  when 
the  fall  of  two  bodies,  commencing  their  falls  at  different  distan- 
ces are  compared,  that  it  becomes  necessary  to  assume  one  of 
the  bodies  at  unity,  and  the  other  at  some  other  distance,  either 
greater  or  less  than  unity,  as  we  may  choose. 

The  comparison,  however,  may  be  made  by  means  of  two 
simple  formulas,  as  in  case  of  the  foregoing ;  the  first,  or  that 
preceding  the  double  line,  being  composed  of  two  figures  only, 
each  denoting  unity,  the  upper  figure  of  which  may  denote  either 
unity  of  distance  from  the  centre  of  gravity,  unity  of  time 
in  which  the  fall  or  operation  is  to  take  place,  or  unity  of  force 
or  gravity,  namely,  the  whole  amount  of  force  or  gravity  expend- 
ed on  the  body  during  the  operation,  namely,  during  its  fall 
over  the  given  space  ;  and  the  lower  figure  in  the  first  formula 
denoting  the  given  space  to  be  described  or  passed  over,  namely, 
a  unit  of  space,  or  the  same  amount  that  is  to  be  described  or 
passed  over  in  the  second  formula,  or  that  succeeding  the  double 
line. 

The  uniform  series  in  the  second  formula  denotes  as  many 
unities  of  distance  from  the  centre  of  gravity,  or  as  many  unities 
of  time  in  which  the  operation  is  to  be  performed,  as  there  are 
figures  in  the  series ;  and  the  whole  amount  of  force  of  gravity 
(whatever  such  amount  may  be)  is  divided  into  as  many  equal 
parts  as  there  are  figures  in  the  uniform  series ;  namely,  into  as 
many  equal  parts  as  there  are  units  of  distance,  or  units  of  time  *r 
and  the  odd  numbers  in  the  lower  series  of  the  second  formula 
are  used  to  divide  the  given  space  to  be  described  (namely,  a 
unit  of  space,  or  the  same  amount  denoted  by  the  lower  figure  in 
the  first  formula)  into  as  many  equal  parts  as  there  are  units  con- 
tained in  all  the  odd  numbers  from  1  to  15  inclusive,  namely, 
64,  or  the  square  of  the  units  of  distance,  or  of  the  time  of  the 
fall,  as  also  of  the  whole  amount  of  force  expended  during  the 
fall,  or,  if  we  please,  of  the  sum  of  the  equal  parts  into  which  the 


126  ON    THE    LAW    OF    GRAVITY. 

whole  amount  of  force  is  divided  by  the  uniform  series ;  and 
then  l-64th  of  the  space  will  be  described  while  l-8th  of  the  force 
is  being  expended;  3-64ths  more  of  the  space  will  be  described 
while  another  l-8th  of  the  force  is  being  expended,  and  so  on  ;  the 
amount  of  space  described  increasing  by  the  odd  numbers,  while 
the  force  is  uniform.  Now  the  intensity  or  rate  of  force  em- 
ployed in  the  second  formula  is  less  by  some  given  ratio  than  that 
employed  in  the  first  formula,  —  that  employed  in  the  first  being 
unity  ;  and  the  ratio  of  the  intensity  of  the  force  employed  in  the 
second  formula,  to  that  employed  in  the  first,  is  what  we  are 
seeking  for,  which,  of  course  will  give  the  ratio  of  the  rate  or  in- 
tensity of  the  force  to  that  of  the  distance  from  the  centre  of 
gravity.  Now  in  either  formula,  the  last  15-64ths  of  the  unity 
of  space  will  be  described,  while  an  equal  amount  of  force  is 
being  expended,  as  while  the  first  l-64th  of  the  space  is  being 
described  ;  for  the  law  is  the  same  in  both  formulas.  But  it  is 
too  serious  a  question  to  be  lightly  determined,  whether  the 
amount  of  force  expended  upon  the  body  in  the  first  formula, 
while  passing  over  the  first  l-64th  of  the  given  space,  or  the  las-t 
15-64ths  of  the  given  space,  shall  be  eight-fold,  or  only  equal  to 
the  amount  of  force  expended  on  the  body  in  the  second  for- 
mula while  passing  over  the  first  l-64th,  or  the  last  15-64ths  of 
the  given  space,  or  unit  of  space  ;  namely,  whether  effective  force 
shall  perform  the  labor,  or  whether  7-8ths  of  the  labor  shall  be  de- 
nied to  the  operation  of  force,  and  credited  to  passive  time ;  thereby 
assigning  or  requiring  the  same  amount  of  force  to  be  expended 
on  the  body  in  the  first  formula  while  describing  the  first  l-64th 
part  of  the  unit  of  space  to  be  described,  as  will  be  expended  on 
the  body  in  the  second  formula,  while  describing  the  whole  of 
the  unit  of  space ;  which  is  readily  perceived  to  be  the  case,  if 
the  rate  of  force  varies  inversely  as  the  square  of  the  distance  ; 
while  if  the  rate  of  force  varies  inversely  as  the  distance,  eight- 
fold the  amount  of  force  will  be  expended  upon  the  body  in  the 
first  formula,  while  describing  the  whole  unit  of  space  that  is  ex- 
pended upon  the  body  in  the  second  formula,  while  describing 
the  first  l-64th,  or  the  last  15-64ths  of  the  whole  unity  of  space ; 
that  is,  in  either  formula,  the  amount  of  space  described  at  any 
given  instant  of  time  during  the  fall  over  the  given  space,  will  be 
the  square  of  the  amount  of  force  then  agd  there  expended  from 
the  commencement  of  the  fall ;  and  in  such  case  passive  time 
will  have  effected  nothing,  or  will  have  had  no  effective  or  phys- 
ical agency  in  the  matter*  any  more  than  space  or  distance  has  ; 
it  will  only  have  awaited  the  operation  of  force  according  to  the 
rate  or  intensity  of  the  force  applied,  precisely  as  in  the  case  of 
the  mechanical  powers,  in  which  case  the  whole  time  recipro- 
cates with  the  rate  or  intensity  of  the  force  employed, 


ON    THE    LAW    OF    GRAVITY.  127 

But  1  will  defer  for  the  present  a  further  investigation  of  the 
law  of  gravity  in  and  through  a  consideration  of  the  law  of  fall- 
ing bodies,  as  it  is  my  intention  to  refer  to  and  examine  this 
very  important  subject  again ;  and  until  I  can  induce  a  consider- 
ation of  the  subject  by  others,  I  wish  to  present  the  subject  in 
various  ways,  (and  they  may  be  multiplied  to  any  extent,)  with 
the  view  to  ascertain  whether  that  great,  eternal  and  universal 
law,  upon  which  we  all  suppose  we  have  a  right  to  calculate, 
(viz.  that  the  whole  amount  of  time  employed  or  rather  passed 
over  during  a  mechanical  operation,  is  the  simple  inverse  or 
reciprocal  of  the  rate  of  force  employed)  has  anywhere,  or  in 
any  case,  been  abrogated,  repealed  or  altered  ;  and  whether  time, 
instead  of  passively  awaiting  an  operation,  has  become  the  prin- 
cipal operator.  For  if  such  be  the  case,  it  is  certainly  as  im- 
portant to  philosophy  that  it  be  known,  as  Mr.  Rees  or  any  other 
person  can  consider  it.  But  surely  such  could  not  have  been 
suspected  previous  to  its  discovery,  or  its  being  found  by  New- 
ton, after  a  search  of  some  twenty  years,  (according  to  his  biog- 
rapher,) and  even  then,  under  such  ecstatic  agony  as  compelled 
him  to  ask  a  friend  to  pick  up  and  secure  what  had  been  found ; 
and  the  importance  of  which  discovery  was  such  as  to  require 
two  years  in  penetrating  its  consequences. 

In  respect  to  the  line  of  direction,  which  a  body  falling  towards 
the  earth  will  form  or  describe,  —  which  was  considered  and  moot- 
ed between  Newton  and  Mr.  Hook,  as  though  the  same  were  to 
be  settled  by  convention,  —  Hook  suggested,  that  if  the  body  fall 
in  a  vacuum,  the  line  would  be  an  eccentric  ellipse ;  and  also 
suggested  such  eccentric  ellipse  as  the  consequence  of  a  force 
inversely  proportional  to  the  squares  of  the  distances  from  the 
earth's  centre.  In  relation  to  this  matter,  it  is  said  that  Newton, 
"  having  examined  this  suggestion  by  mathematical  calculations, 
found  that  an  attractive  force,  emanating  from  a  centre,  and  act- 
ing inversely  as  the  squares  of  the  distances,  would  produce 
motions  exactly  resembling  planetary  motions,  both  in  regard  to 
the  form  of  the  orbit  and  the  velocity  of  the  body  at  each  point;" 
and  this,  he  says,  "  was  the  secret  of  the  system  of  the  world  ; 
but  it  still  remained  to  account  for  the  discordance  of  t^ie  moon's 
motion,  which  had  before  (1665)  embarrassed  Newton."  Now 
what  Sir  Isaac  Newton  is  said  to  have  found  by  such  mathe- 
matical calculations,  has  been  quite  liberally  endorsed  to  the 
world  as  a  demonstrated  truth.  But  I  predict  that  the  world 
(sooner  or  later)  will  esteem  it  as  great  an  imposition  as  could 
well  have  been  palmed  off  upon  it.  Thus  Mr.  Rees,  in  order  to 
help  Newton's  moon  story  out  of  the  fog  in  which  it  is  envel- 
oped, says,  that  the  centripetal  force  of  the  moon  is  known  from 


128 


ON    THE    LAW    OF    GRAVITY. 


the  elliptical  figure  of  the  moon's  orbit  to  be  reciprocally  propor- 
tional to  the  square  of  the  distance. 

Nevertheless,  even  after  this  supposed  mathematical  discovery 
in  respect  to  Hook's  suggestion,  the  motions  of  the  moon  were 
still  discordant  until  they  were  reconciled  to  the  same  mathe- 
matical disclosure  by  a  deduction  of  the  Newtonian  law  of 
gravity,  from  a  consideration  of  the  law  of  falling  bodies,  which 
was  almost  impenetrable  in  its  consequences,  and  laid  the  foun- 
dation for  the  immortal  "  Principia." 

We  see,  then,  what  supposed  results  are  attempted  to  be 
linked  together,  or  placed  in  the  same  category,  however  discord- 
ant, to  induce  a  belief  that  the  force  of  gravity  varies  inversely 
as  the  square  of  the  distance  varies ;  and  that  the  same  has  been 
so  found  or  demonstrated  by  Sir  Isaac  Newton,  —  the  most  con- 
clusive method,  and  that  of  which  the  modus  operandi  is  given, 
(finished  out  by  a  friend  of  Newton,)  being  the  one  arising  from 
a  consideration  of  the  law  of  falling  bodies. 

The  law  of  falling  bodies,  then,  from  which  the  law  of  gravity 
may  be  easily  deduced,  and  in  which  no  error  or  mistake  in  the 
course  of  the  investigation  need  to  occur,  (unless  it  be  in  assign- 
ing some  active  operation  to  time,)  ought,  again  and  again,  to  be 
investigated.  I  am  aware  that  time  has  been  so  far  personified, 
as  to  make  it  the  principal  actor  in  our  world,  and  even  in  the 
universe.  We  speak  of  the  hand,  the  tooth,  the  foot,  the  track 
of  time ;  of  its  ravages,  of  the  work  of  time,  &c. ;  we  speak  of 
beguiling  time,  and  even  of  killing  time.  Such  personification, 
however,  is  not  poetic  enough  to  give  us  a  clear  idea  of  the 
species  of  animated  nature  to  which  time  belongs ;  but  when 
we  find  time  seated  at  the  festive  board,  with  ermine  beard  and 
forelock  gray,  having  laid  down  his  scythe  and  glass,  and  be- 
come a  cheerful  spectator  of  the  proceedings,  we  can  have  but 
little  doubt  of  its  human  nature,  even  though  it  should  not  oth- 
erwise partake  of  the  festivities. 

Nevertheless,  after  all  this  personification  of  time,  in  matters  of 
philosophy,  we  ought  not  to  assign  to  time  a  task,  a  labor,  or  an 
operation  which  it  is  wholly  incompetent  to  perform. 

\ 

SECTION    FOURTH. 

In  an  investigation  of  the  law  of  gravity  by  the  use  of  the 
means  which  have  usually  been  resorted  to  by  those  who  have 
attempted  the  investigation,  viz.  a  consideration  of  the  law  of 
falling  bodies,  it  may  not  be  amiss,  as  preliminary  thereto,  to 
quote  from  those  authorities  which  may  be  supposed  to  contain 
the  best  expositions  of  the  Newtonian  method  of  investigation. 


ON    THE    LAW    OF    GRAVITY.     -  129 

Colin  Maclaurin,  who  was  a  celebrated  mathematician,  a  con- 
temporary of  Sir  Isaac  Newton,  and  an  admirer  of  his  philoso- 
phy and  discoveries,  and  who  wrote  an  account  of  them  in  four 
books,  which  passed  through  several  editions,  thus  gives  us  the 
Newtonian  exposition  or  deduction  of  the  law  of  gravity  from  a 
consideration  of  the  law  of  falling  bodies.  "  The  computation 
may  be  made  in  this  manner,  —  the  mean  distance  of  the  moon 
from  the  earth  being  sixty  times  the  distance  of  heavy  bodies  at 
the  surface  from  its  centre,  (a  body  at  the  surface  will  fall 
15xV  Parisian  feet  in  one  second,)  and  her  gravity  increasing 
in  proportion  as  the  square  of  the  distance  from  the  centre  of  the 
earth  decreases,  her  gravity  would  be  60  X  60  times  greater  near 
the  surface  of  the  earth  than  at  her  mean  distance,  and  therefore 
would  carry  her  through  60  X  60  X  15TV  Parisian  feet  in  a 
minute,  near  the  surface  ;  but  the  same  power  would  carry  her 
through  60  X  60  times  less  space  in  a  second,  than  in  a  minute, 
by  what  has  been  often  observed  of  the  descent  of  falling  bodies ; 
and  therefore  the  moon,  in  a  second  of  time,  would  fall  by  her 
gravity  near  the  surface  of  the  earth  15  TV  Parisian  feet,  which 
therefore  is  the  same  with  the  gravity  of  terrestrial  bodies." 
I  will  also  refer  to  Rees'  Cyclopaedia,  Vol.  XXV,  under  the 
article  Moon,  in  which  the  subject  under  consideration  is  treated 
of;  when,  after  a  comparison  of  the  sun,  moon,  and  earth,  as  to 
density,  proportional  magnitude,  &c.,  and  after  some  remarks  on 
the  motion  of  the  moon's  apogee,  its  irregularities,  &c.,  we  find 
the  following  observation,  viz.,  "  The  irregularities  above  enu- 
merated, and  some  others  of  a  similar  kind,  have  been  urged  as 
objections  to  the  Newtonian  theory  of  gravity,  though  they  were 
anticipated  by  the  illustrious  author,  who  not  only  evinced  their 
consistency  with  it,  but  suggested  the  explication  of  them  which 
might  be  deduced  from  that  theory  properly  understood  and  ap- 
plied." Mr.  Rees  then  quotes  from  Mr.  Vince,  commencing 
with  the  familiar  expression,  "  Sir  Isaac  Newton  having  found," 
&c.,  and  finally  observes,  that  "  Mr.  Euler  also  retracted  his  own 
erroneous  opinion  in  deference  to  the  judgment  of  Mr.  Clairaut, 
and  concurred  with  him  in  doing  ample  justice  to  the  Newtonian 
theory,"  &c.  Mr.  Rees  proceeds  to  say,  that  "  It  would  not  be 
consistent  with  the  limits  or  nature  of  this  work,  [the  Cyclopae- 
dia,] to  investigate,  by  tedious  an,d  elaborate  processes  of  an 
analytical  and  geometrical  kind,  the  various  equations  that  have 
been  explored  for  the  illustration  of  these  laws,"  and  refers  the 
reader  to  Vince's  Complete  System  of  Astronomy, '  Vol.  II. 
Chap.  XXXII.  Mr.  Rees  then  proceeds:  "We  shall,  however, 
in  this  place,  introduce  a  general  view  of  the  Newtonian  theory 
of  gravity,  as  it  i?  applied  to  the  solution  of  the  irregularities  of 


130  ON    THE   LAW    OF   GRAVITY. 

the  moon's  motions."  "We  have  already,  under  the  article 
Gravitation,  illustrated  and  confirmed  the  Newtonian  theory  of 
gravity,  as  it  regards  the  moon  and  other  planets ;  but  as  the 
subject  is  of  importance,  and  as  it  is  immediately  connected 
with  what  follows,  we  shall  here  give  a  concise  statement  of  the 
leading  facts  by  which  the  identity  of  the  centripetal  force  as  it 
respects  the  moon  and  that  of  gravity,  was  originally  explained 
and  established,  referring,  for  a  more  detailed  account,  to  the 
article  just  cited." 

"  It  is  well  known  and  universally  allowed,  that  the  planets 
are  retained  in  their  orbits  by  some  power  which  is  constantly 
acting  upon  them  ;  that  this  power  is  directed  towards  the  cen- 
tres of  their  orbits  ;  that  the  efficacy  of  this  power  increases  upon 
an  approach  to  the  centre,  and  diminishes  by  its  recess  from  the 
same ;  and  that  it  increases  according  to  a  certain  law,  namely, 
that  of  the  square  of  the  distance  as  the  distance  diminishes ; 
and  it  diminishes  in  the  same  ratio  as  the  distance  increases. 
Now  by  comparing  this  centripetal  force  of  the  planets  with  the 
force  of  gravity  on  the  earth,  they  will  be  found  perfectly  alike. 
This  we  shall  illustrate  in  case  of  the  moon,  the  nearest  to  us  of 
all  the  planets."  "  The  rectilinear  spaces  described  in  any  given 
time  by  a  falling  body,  urged  by  any  powers,  reckoning  from 
the  beginning  of  its  descent,  are  proportionable  to  those  powers. 
Consequently  the  centripetal  force  of  the  moon,  revolving  in  its 
orbit,  is  to  the  force  of  gravity  on  the  surface  of  the  earth,  as  the 
space  which  the  moon  would  describe  in  falling  any  little  timey 
by  her  centripetal  force  towards  the  earth,  if  she  had  no  circular 
motion  at  all,  —  is  to  the  space  which  a  body  near  the  earth  would 
describe  in  falling  by  its  gravity  towards  the  same."  "  By  a  very 
easy  and  obvious  calculation  of  these  two  spaces,  it  will  appear 
that  the  first  of  them  is  to  the  second,  —  that  is,  the  centripetal  force 
of  the  moon  revolving  in  her  orbit  is  to  the  force  of  gravity  oil 
the  surface  of  the  earth,  —  as  the  square  of  the  earth^s  semi-diame- 
ter is  to  the  square  of  the  semi-diameter  of  her  orbit,  which  is 
the  same  ratio  as  the  moon's  centripetal  force  in  her  orbit  is  to 
the  same  force  near  the  surface  of  the  earth.  The  moon's  cen- 
tripetal force  is  therefore  equal  to  the  force  of  gravity.  These 
forces  consequently  are  not  different,  but  they  are  one  and  the 
same ;  for  if  they  were  different,  bodies  acted  on  by  the  two 
powers  conjointly  would  fall  towards  the  earth  with  a  velocity 
double  that  arising  from  the  sole  power  of  gravity.  It  is  evi- 
dent, therefore,  that  the  moon's  centripetal  force,  by  which  she  is 
retained  in  her  orbit  and  prevented  from  running  off  in  a  tangent, 
is  the  very  power  of  gravity  of  the  earth  extended  thither."  Mr. 
Rees  then  proceeds  to  give  the  method  of  investigation  after  the 


ON   THE    LAW    OP    GRAVITY.  131 

manner  of  Maclaurin,  and  also  of  Mr.  Vince,  namely,  that  Sir 
Isaac  Newton  found  that  the  moon  is  drawn  by  its  centripetal 
force  from  a  tangent  to  its  orbij;  towards  the  centre  of  the  earth, 
about  15fV  Parisian  feet  in  one  minute  of  time ;  and  that  a 
body  will  fall  from  a  state  of  rest  near  the  surface  of  the  earth 
15y^  Parisian  feet  in  a  second  of  time ;  and  that  the  distance 
of  the  moon  from  the  centre  of  the  earth  is  about  sixty  times  as 
great  as  the  surface  of  the  earth  is  from  its  centre,  &c.  Mr.  Rees 
then  says  :  "  But  this  force  (centripetal  force  of  the  moon)  being 
known  from  the  elliptical  figure  of  her  orbit  to  be  reciprocally 
proportioned  to  the  square  of  the  distance,  would  impel  the  moon, 
supposed  to  be  at  the  surface  of  the  earth,  through  a  space  equal 
to  60  X  60  X  15^  feet  in  one  minute."  "  But  bodies  impel- 
led by  the  force  of  gravity  fall,  near  the  surface  of  the  earth, 
through  the  space  of  15TV  Paris  feet  in  one  second,  and  the 
spaces  being  as  the  squares  of  the  times,  through  60X60X15-^ 
in  a  minute.  Consequently,  as  the  force  by  which  the  moon  is 
retained  in  its  orbit,  and  the  force  of  gravity,  produce  the  same 
effects  in  the  same  circumstances,  and  tend  towards  the  same 
point,  they  are  the  same  force."  Mr.  Vince,  who  seems  to  have 
been  taught  according  to  the  perfect  manner  of  the  law  of  the 
fathers,  and  who  appears  to  be  a  very  honest  recorder  of  the  phi- 
losophy extant,  from  some  cause  seems  to  have  avoided  so  de- 
tailed an  exposition  of  the  Newtonian  method  of  deducing  the 
law  of  gravity,  as  some  others  have  presented  ;  and  in  place  of  it 
find  the  following,  namely:  "  Sir  Isaac  Newton  found  that  if 
we  the  force  with  which  bodies  fall  upon  the  earth's  surface  were 
extended  to  the  moon,  and  were  to  vary  inversely  as  the  square 
of  the  distance  from  the  centre  of  the  earth,  it.  would  in  one  min- 
ute draw  the  moon  through  a  space  which  is  equal  to  the  versed 
sine  of  the  arc  which  the  moon  describes  in  one  minute.  He 
concluded,  therefore,  that  the  moon  was  retained  in  its  orbit  by 
the  same  force  as  that  by  which  bodies  are  attracted  upon  the 
earth." 

I  have  thought  proper  to  quote  thus  liberally  from  those  text 
books  which  have  been  received  as  authority,  (upon  a  subject 
which  Mr.  Rees  considers  important,)  from  the  time  of  the  dis- 
pensation of  the  law  by  Sir  Isaac  Newton,  until  it  became  whol- 
ly unnecessary  to  do  more  than  simply  to  declare  the  law,  name- 
ly, "  that  the  force  of  gravity  varies  inversely  as  the  square  of  the 
distance  varies." 

The  subject  is  not  less  important  than  it  is  considered  to  be 
by  Mr.  Rees,  and  consequently  is  entitled  to  as  much  consider- 
ation as  will  serve  to  disclose  a  truth  so  important  to  be  known. 
The  subject  is  not  a  new  one,  requiring  only  a  simple  disclosure 


132  ON    THE    LAW    OF    GRAVITY. 

and  development  to  the  world,  but  as  it  has  been  husbanded  and 
fortified  about  by  eminent  tacticians  and  engineers,  until  at 
length  the  world  have  assented  that  the  premises  are  legitimately 
and  rightfully  possessed,  (especially  after  the  adhesion  to  them 
of  such  men  as  Clairaut  and  Euler,)  something  more  must  be 
done  than  simply  to  disclose  the  truth  to  unprejudiced  minds. 
Error  must  be  broken  up  and  dispersed  ;  a  breach  and  havoc 
must  be  made  in  the  citadel  ;  the  old  standard  must  be  torn 
down  and  a  new  one  firmly  placed  in  its  stead,  with  legible  in- 
scriptions thereon  which  all  may  read ;  and  should  such  stand- 
ard appear  more  republican  or  less  regal  than  the  old,  it  may 
only  indicate  one  step  in  the  march  of  modern  improvement. 
Hence  it  may  not  be  inappropriate,  as  preliminary  to  a  more  di- 
rect investigation  of  the  subject,  that  I  should  remark  or  com- 
ment somewhat  upon  the  text  just  quoted  from  Maclaurin,  Rees 
and  Vince ;  and  also  upon  other  extracts  which  may  be  taken  or 
considered  in  connection.  Saying  nothing  about  the  discrepancies 
which  may  appear  in  these  quotations  when  taken  in  connec- 
tion, I  shall  take  the  liberty  to  discard  from  the  investigation  or 
comments,  that  barren,  absurd  and  futile  allegation  which  inex- 
plicability  required  as  a  make-weight,  and  which  in  process  of 
time  was  so  opportunely  added ;  and  shall  leave  the  same  to  be 
inquired  of  by  the  world  at  their  leisure,  —  I  refer  to  that  allega- 
tion of  Mr.  Rees,  (which,  to  be  sure,  seems  rather  in  the  nature 
of  an  exclusion  of  a  conclusion,)  that  "  this  force  [the  centripetal 
force  of  the  moon]  is  known,  from  the  elliptic  figure  of  her  orbit, 
to  be  reciprocally  proportional  to  the  square  of  the  distance,"  &c., 
which  naked  allegation  has  been  adopted  as  law,  as  in  case  of 
Norton's  Astronomy,  "  designed  for  use  as  a  text-book  in  col- 
leges and  academies  ;  "  in  which,  under  the  head  of  the  "  theory 
of  universal  gravitation,"  he  says,  "  It  is  also  proved  by  the  prin- 
ciples of  mechanics,  that  if  a  body,  continually  urged  by  a  force 
directed  to  some  point,  describe  an  ellipse  of  which  that  point  is 
a  focus,  the  force  by  which  it  is  urged  must  vary  inversely  as  the 
square  of  the  distance."  Now  allegations  like  this  may  be  ne- 
cessary for  the  establishing  of  the  Newtonian  law  of  gravity,  if 
found  to  be  inexplicable  without  them  ;  but  they  are  not  neces- 
sary for  the  furtherance  of  philosophy,  or  the  science  of  astrono- 
my. If  the  orbits  are  elliptic,  and  if  in  an  elliptic  orbit  the  cen- 
tripetal force  is  necessarily  inversely  as  the  square  of  the  distance, 
and  if  this  were  proved  to  be  the  case,  then,  to  be  sure,  Rees'  alle- 
gation would  be  well  enough,  because  true  ;  but  as  it  is,  the  very 
fact  is  assumed,  a  priori,  for  which  the  inquiry  was  being  made. 
An  ellipse  may  vary  from  the  circle  by  any  given  rate  of  ellipti- 
city  ;  consequently  the  variation  may  be  less  than  any  assignable 


ON    THE    LAW    OF    GRAVITY.  133 

quantity ;  and  consequently  we  all  agree,  that  if  the  force  of  grav- 
ity be  inversely  as  the  distance  in  the  ellipse,  it  will  also  be  so  in 
the  circle;  and,  e  converso,  if  the  .force  be  inversely  as  the  dis- 
tance in  the  circle,  it  will  be  also  in  the  ellipse,  if  a  planet  shall 
be  found  revolving  in  such  orbit.  But  the  simple  laws  and  prin- 
ciples of  dynamics,  so  readily  evolved  and  explained,  are  driven 
or  warped  from  their  legitimate  sphere,  for  the  purpose  of  estab- 
lishing belief  in  a  hypothesis  which  yet  requires  investigation  to 
determine  its  truth  or  error.  It  cannot  be  necessary  that  heaven 
and  earth  should  fail,  rather  than  that  Sir  Isaac  Newton  should 
be  found  to  have  been  mistaken,  either  in  his  theory  or  law  of 
gravity. 

As  a  prelude  to  the  commentaries  intended  upon  the  quota- 
tions before  made,  and  also  to  such  further  matter  as  I  may  put 
forth  for  consideration,  perhaps  no  better  can  be  offered  than  an  ex- 
tract from  Colin  Maclaurin,  (the  friend  and  admirer  of  Sir  Isaac 
Newton  and  his  philosophy,)  in  which  he  sets  forth  the  tribula- 
tions of  science  in  most  appropriate  remarks,  as  in  the  following : 

"  An  entire  liberty  must  be  allowed  to  our  inquiries,  that  natu- 
ral philosophy  may  become  subservient  to  the  most  valuable  pur- 
poses, and  acquire  all  the  certainty  and  perfection  of  which  it  is 
capable  ;  but  we  ought  not  to  abuse  this  liberty  by  supposing 
instead  of  inquiring,  and  by  imagining  systems  instead  of  learn- 
ing from  observation  and  experience  the  true  construction  of 
things.  Speculative  men,  by  the  force  of  genius,  may  invent 
systems  that  perhaps  will  be  greatly  admired  for  a  time.  These, 
however,  are  phantoms  which  the  force  of  truth  will  sooner  or 
later  dispel,  and  while  we  are  pleased  with  the  deceit,  true  phi- 
losophy, with  all  the  arts  and  improvements  that  depend  upon  it, 
suffers." 

"  The  real  state  of  things  escapes  our  observation,  or  if  it  pre- 
sents itself  to  us,  we  are  apt  either  to  neglect  it  wholly  as  fiction, 
or,  by  new  efforts  of  a  vain  ingenuity,  to  interweave  it  with 
our  own  conceits,  and  labor  to  make  it  tally  with  our  favorite 
schemes.  Thus  by  blending  together  parts  so  ill  suited,  the 
whole  comes  forth  an  absurd  composition  of  truth  and  error." 
"  Of  the  many  difficulties  that  have  stood  in  the  way  of  philoso- 
phy, this  vanity  perhaps  has  had  the  worst  effects,"  &c.  "  All  oth- 
er causes  have  not  done  so  much  harm  as  that  pride  and  ambi- 
tion which  have  led  philosophers  to  think  it  beneath  them  to  offer 
anything  less  to  the  world  than  a  complete  and  finished  system 
of  nature ;  and  in  order  to  obtain  this  at  once,  to  take  the  liberty 
of  inventing  certain  principles  and  hypotheses,  from  which  they 
pretended  to  explain  all  her  mysteries."  Mr.  Maclaurin,  how- 
ever, in  speaking  of  Sir  Isaac  Newton's  philosophy,  evinces  no 


134  ON    THE    LAW    OF    GRAVITY. 

want  of  confidence  in  its  perfection  ;  for  he  says  that  Sir  Isaac 
Newton  "  has  secured  his  philosophy  against  any  hazard  of  being 
disproved  or  weakened  by  future  discoveries."  Mr,  Maclaurin, 
however,  laments  bitterly,  approximating  even  to  poetic  agony, 
that  the  Newtonian  philosophy  should  have  been  so  long  resist- 
ed and  delayed.  He  says,  "  It  was,  however,  no  new  thing  that 
this  philosophy  should  meet  with  opposition,  &c.,  and  that  it 
should  struggle  with  the  prejudices  of  those  who  had  accustom- 
ed themselves  to  think  only  in  a  certain  systematic  way  ;  who 
could  not  be  prevailed  upon  to  abandon  their  favorite  schemes, 
while  they  were  able  to  imagine  the  least  pretext  for  continuing 
the  dispute  ;  every  art  and  talent  were  displayed  to  support  their 
falling  cause ;  no  aid  seemed  foreign  to  them  that  could  in  any 
manner  annoy  their  adversary;  and  such  often  was  their  obsti- 
nacy, that  truth  was  able  to  make  but  little  progress,  till  they 
were  succeeded  by  younger  persons,  who  had  not  so  strongly 
imbibed  their  prejudices. " 

Mr.  Maclaurin  having  fully  imbibed  the  Newtonian  theory 
and  law  of  gravity,  and  also  his  projectile  hypothesis,  namely, 
that  the  planets  must  of  necessity  have  originally  been  projected 
in  tangents  to  their  orbits,  with  forces  so  proportioned  to  their 
powers  of  innate  or  inherent  gravity,  as  to  enable  such  innate 
gravity  to  form  the  orbit,  either  more  or  less  eccentric,  according  to 
the  original  force  of  projection,  (which  he  calls  "  the  velocity 
of  projection,")  —  then  labors  to  show  how  a  planet  will  proceed 
in  its  orbit, —  what  kind  of  curve  it  will  form  when  projected  in 
a  tangent  to  its  orbit,  with  a  greater  or  less  force,  at  a  greater  or 
less  distance  from  the  sun,  —  what  kind  of  an  ellipse  it  would 
form  with  a  greater  or  less  projection,  which  philosophy  he  has  em- 
bellished and  elucidated  by  a  multiplicity  of  diagrams,  which 

"  Gird  the  sphere  with  centric  and  eccentric,  scribbled  o'er  " 

with  all  the  fanciful  notions  in  regard  to  the  manner  in  which 
bodies  would  proceed,  —  what  curves  they  would  form,  —  how 
the  force  and  the  motion  would  alternately  get  the  better  of  each 
other,  — •  how  a  body  might  forever  continue  to  recede  from  the 
sun ;  or  upon  the  other  hand,  how  it  might  directly  or  eventually 
fall  into  the  sun,  &c.,  were  the  laws  of  nature  which  govern  the 
universe,  altogether  different  from  what  they  are; — together  with 
a  vast  amount  of  similar  matter,  well  calculated  to  bewilder  the 
mind  into  the  belief  that  the  force  of  gravity  is  inversely  as  the 
square  of  the  distance,  —  that  the  planets  were  necessarily  pro- 
jected in  tangents  to  their  orbits;  and  hence,  —  that  a  planet  may 
revolve  in  an  eccentric  orbit,  and  still  obey  the  Newtonian  law 
of  gravity,  —  it  is  necessary  to  attach  a  kind  of  balance-wheel, 


ON   THE    LAW    OF    GRAVITY.  135 

called  the  centrifugal  force,  to  enable  the  force  and  motion  alter- 
nately to  get  the  better  of  each  other  at  the  proper  times  and 
places.  And  in  respect  to  the  centrifugal  force,  I  will  here  give 
a  quotation  from  the  author.  "  The  gravity  indeed  at  the  lower 
apside,  is  greater  than  the  gravity  at  the  higher  apside,  in  pro- 
portion as  the  square  of  the  distance  is  less.  But  the  centrifu- 
gal force  arising  from  the  circular  motion  about  the  sun,  increases 
in  a  higher  proportion,  namely,  as  the  cubes  of  the  distances 
decrease,  for  these  centrifugal  forces  are  in  the  direct  proportion 
to  the  squares  of  the  velocities,  and  their  inversed  proportion  of 
the  distances  compounded  together ;  the  first  of  these  in  the 
inverse  proportion  of  the  squares  of  the  distances,  and  the  two 
together  compound  the  inverse  proportion  of  the  cubes  of  the 
distances.  The  centrifugal  force  therefore  increases  in  a  more 
rapid  ratio  than  the  force  of  gravity  ;  and  though  the  force  of 
gravity  prevails  in  the  higher  part  of  the  orbit,  the  centrifugal 
force  gets  the  better  in  the  lower  part  of  it.  The  force  of 
gravity  prevailing  in  the  higher  apsis,  makes  the  planet  approach 
the  sun,  the  centrifugal  force  prevailing  in  the  lower  apsis  makes 
the  body  recede  from  the  sun ;  and  the  continued  action  of  these 
two  forces  causes  the  planet  forever  to  revolve  from  one  to  the 
other." 

Hence,  if  we  are  to  believe  Mr.  Maclaurin,  the  burden  of  the 
Newtonian  philosophy  in  respect  to  the  law  of  force  and  motion 
by  which  a  planet  revolves  in  its  orbit,  is,  to  use  the  very  language 
or  expression  of  Maclaurin,  "  that  the  force  and  motion  alternately 
get  the  better  of  each  other."  This  then  is  the  only  possible 
mode  devised  for  compelling  the  planets  to  obey  the  New- 
tonian law  of  gravity,  as  promulgated  ;  for  even  Mr.  Maclaurin, 
had  there  been  "an  entire  liberty  allowed  to  his  inquiries," 
(which  he  so  well  recommended  in  the  outset,)  could  have  de- 
vised a  much  better  law ;  for  after  all  his  labors,  wanderings  and 
perplexities,  to  which  he  was  doomed  in  consequence  of  having 
implicitly  imbibed  the  Newtonian  law  of  gravity,  we  find  him 
alleging  and  even  repeating  the  allegation,  that  the  proper  limit 
of  gravitation,  by  which  the  planets  could  best  revolve,  would 
be,  "  the  inverse  simple  proportion  of  the  distance,"  and  he  at- 
tempts to  show  how  beautifully  a  planet  would  revolve  from  one 
apsis  to  the  other,  if  operated  upon  by  a  force  of  gravity,  de- 
creasing in  the  reciprocal  simple  proportion  of  the  distance. 

I  have  thus  far  quoted  from,  and  referred  to  the  works  of  Mr. 
Maclaurin,  with  a  view  to  show  what  the  obstinate  world  had 
to  contend  with,  inasmuch  as  he  was  contemporary  with  Sir  Isaac 
Newton,  but  long  survived  him ;  and  as  he  became  the  able 
expounder  of  the  Newtonian  philosophy  to  the  world,  and 
especially  of  the  modus  operandi  by  which  the  laws  of  planetary 


136  ON    THE    LAW    OF    GRAVITY. 

force  and  motion  were  to  operate  in  accordance  with  the  New- 
tonian theory  and  law  of  gravity,  it  might  naturally  be  sup- 
posed that  through  Maclaurin,  we  should  drink  from  the  highest 
and  purest  streams  of  the  Newtonian  philosophy.  But  the 
inquirer  had  better  examine  the  work  if  it  be  accessible. 

Gravity  may  be  technically  denned  by  saying  that  it  is  the 
force  which  urges  a  planet  towards  the  sun,  and  this  force  is 
called  the  gravity  of  the  planet,  or  its  force  of  gravity  towards 
the  sun ;  and  in  case  of  a  body  falling  towards  the  earth  from 
a  state  of  rest,  the  motion  of  which  is  uniformly  accelerated  from 
the  beginning  or  commencement  of  the  fall,  —  there  being  no 
space  however  small,  without  acceleration,  —  the  force  which 
constantly  accelerates  the  motion  may  be  called  either  the  ac- 
celerating force,  the  soliciting  force,  or  the  attractive  force,  or 
force  of  gravity. 

Now  the  specific  manner  in  which  the  computation  is  made, 
whereby  the  Newtonian  law  of  gravity  is  supposed  to  have  been 
deduced  from  a  consideration  of  the  law  of  falling  bodies,  as 
reported  by  Mr.  Maclaurin  and  by  Mr.  Rees,  is  substantially  the 
same,  and  therefore  requires  the  same  comment;  nevertheless 
Mr.  Rees  has  liberalized  somewhat  in  his  preparation,  which 
may  also  require  a  passing  remark ;  but  from  some  peculiar  am- 
biguities in  the  version  given  by  Mr.  Vince  the  same  may  require 
a  somewhat  different  consideration.  By  Mr.  Maclaurin  and  Mr. 
Rees,  the  unit  of  distance  from  the  centre  of  gravity  is  placed  at 
the  surface  of  the  earth,  or  the  semi-diameter  of  the  earth  from 
the  centre  of  gravity  ;  while  the  moon  is  at  the  distance  60,  or  60 
semi-diameters  of  the  earth  from  its  centre.  The  unit  space  to 
be  passed  over  at  the  unit  distance  is  15TV  Parisian  feet,  and  is 
properly  denoted  by  unity  or  1,  whether  it  be  one  rod,  or  more  or 
less  than  one  rod.  And  the  unit  of  time  in  which  the  unit  of 
space  is  to  be  described  at  the  unit  distance,  is  one  second  of 
time.  At  the  distance  60  the  unit  of  space  will  be  described  in 
60  units  of  time. 

'  And  from  these  premises  it  is  required  to  ascertain  and  assign 
the  proportion  or  rate  of  gravity  at  the  distance  60,  to  that  at  the 
distance  1.  Hence  the  rate  of  gravily  or  the  force  of  gravity  at 
the  unit  distance,  or  distance  1,  must  also  be  assumed  at  unity; 
for  force  is  also  quantity,  and  may  as  well  have  its  quantity  nu- 
merically assumed  as  any  other  quantity,  — as  that  of  distance, 
space  or  time.  Nor  is  there  any  other  rational  or  proper  way  for 
obtaining  the  proportion,  rate  or  intensity  of  gravity  at  the  dis- 
tance of  60,  to  that  at  the  distance  1,  than  that  of  assuming  the 
rate  at  the  distance  1,  at  unity,  —  by  which  that  at  the  distance  60 
is  to  be  compared,  —  as  will  be  readily  seen  by  any  one  on  a  little 


ON    THE    LAW    OF    GRAVITY.     *  137 

consideration  of  the  economy  of  numbers,  their  use  and  perfect 
adaptation  in  the  expression  of  quantity  of  whatever  denomin- 
ation, and  more  especially  in  that  transcendental  property  of 
ratio  and  proportion,  which  is  only  developed  by  the  powers  and 
roots  of  numbers,  (the  centre  of  all  ratio  and  proportion  being  in 
unity ;)  and  hence  all  quantities  which  may  be  numerically  ex- 
pressed, whether  of  the  same  or  of  different  denominations, 
may  be  proportioned  to  each  other  and  to  unity,  that  is,  may 
have  their  ratios  numerically  assigned.  The  proper  elements, 
then,  for  determining  the  law  of  gravity  from  the  law  of  falling 
bodies,  are,  —  distance  from  the  centre  of  gravity,  time,  and  space ; 
namely,  the  time  in  which  a  given  operation  is  to  be  performed, 
and  the  space  to  be  described  or  passed  over  in  a  given  time ; 
and  force,  or  the  force  of  gravity,  which  is  the  agent,  the  cause, 
the  physical  power,  which  is  to  perform  the  labor  or  operation ; 
which  is  to  urge  the  body  over  a  given  space  in  a  given  time. 
Force  then  is  the  active  or  efficient  agent  or  element,  in  the  oper- 
ation ;  while  the  time  and  space  passed  over,  (I  will  not  say  the 
time  employed,  or  the  time  spent,)  are  as  passive  as  is  the  dis- 
tance. And  if  we  have  so  far  and  so  long  been  in  the  habit  of 
personifying  time  as  being  a  physical  agent  or  operator,  that  we 
cannot  divest  ourselves  of  that  form  of  thought,  we  had  better 
avoid  the  error  by  supposing  or  imagining  that  time  waits  for 
the  operation  to  be  performed,  or  for  the  labor  to  be  accomplish- 
ed, that  is  the  labor  of  urging  a  ponderous  body  over  a  given 
space ;  and  by  such  consideration  we  shall  find  ourselves  sus- 
tained, corroborated  and  supported,  by  every  principle  of  me- 
chanics or  dynamics  with  which  we  are  acquainted,  or  which 
has  ever  been  disclosed.  The  rate,  intensity  or  efficiency  of 
the  force  employed  or  required  in  raising  a  ponderous  body 
over  a  given  space  is  inversely  or  reciprocally  as  the  whole  time 
passed  over  during  the  operation.  Every  provident  man  pays 
his  hireling  for  the  efficient  labor  performed,  rather  than  for  the 
time  passed  over.  Nevertheless,  if  the  Newtonian  law  of  gravi- 
ty is  true,  no  proposition  can  be  plainer  or  more  easily  compre- 
hended, than  that  fifty-nine  sixtieths  of  the  unit  of  space 
(namely,  of  the  15TV  Parisian  feet)  passed  over  or  described  a 
the  distance  of  the  moon,  or  distance  60,  is  wholly  the  operation 
or  the  effect  of  time  ;  that  is,  but  one  sixtieth  part  of  the  physical 
or  effective  force  will  be  expended  on  the  body  at  the  distance 
60,  in  urging  it  over  the  unit  of  space  that  will  be  expended  on 
the  body  at  the  distance  of  1,  in  urging  it  over  the  same  amount 
of  space,  namely,  the  unit  of  space,  or  the  15yV  Parisian  feet. 

I  have  said  that  distance,  time,  space,  and  the  rate  of  gravity, 
were  the  proper  elements  from  which  to  deduce  the  law  of  grav- 
18 


138  ON    THE    LAW    OF    GRAVITY. 

ity,  from  the  law  of  falling  bodies  ;  and  that  the  quantity  of  time, 
and  space,  and  the  rate  of  gravity,  must  be  assumed  at  unity,  for 
that  body  which  is  to  commence  its  fall  at  the  distance  of  unity  ; 
our  object  being  to  find  the  relative  or  proportional  efficiency  or 
intensity  of  the  physical  agent  or  operator  that  is  to  perform  the 
operation  in  proportion  to  the  distance.  For,  although  the  pro- 
portions of  the  rates  of  velocity  of  the  two  bodies  (conceiving 
them  to  commence  their  fall  at  the  same  instant  of  time,) 
are  to  be  assigned,  and  consequently  the  rate  of  the  velocity  of 
the  body  commencing  its  fall  at  the  distance  of  unity,  at  any 
given  instant  of  time  during  the  joint  fall,  or  at  any  point  of  space 
equi-distant  from  the  point  from  which  they  respectively  com- 
mence their  fall,  is  to  be  assumed  at  unity,  that  the  rate  of  veloc- 
ity of  the  other  may  be  assigned  or  proportioned  to  unity  ; 
nevertheless,  this  proportion  of  velocity  at  any  given  instant  of 
time,  and  consequently  at  any  point  of  equal  space  described  by 
both  bodies  from  the  commencement,  is  a  deduction  which 
readily  follows  and  discloses  itself  from  a  proper  consideration 
of  the  other  elements,  —  namely,  distance,  time,  space,  and  the 
rate  of  gravity.  Nor  has  there  been  error  in  properly  assigning 
this  rate  of  velocity,  (calling  that  unity  at  the  distance  of  unity,) 
for  all  agree  that  the  body  at  the  distance  1,  starts  with  a  velocity 
3600  times  as  great  as  does  the  body  at  the  distance  60;  that 
when  the  body  at  the  distance  1,  has  passed  over  half  a  second 
of  time,  its  velocity  will  be  3600  times  as  great,  as  that  at  the 
distance  60,  when  the  body  has  passed  over  half  a  second  of 
time  ;  that  when  the  body  at  the  distance  1  has  passed  over  one 
foot  of  space,  its  velocity  will  be  3600  times  as  great  as  that  at  the 
distance  60,  when  the  body  has  passed  over  one  foot  of  space,  &c. 
We  then  speak  properly,  when  we  say  the  rate  of  the  velocity 
is  inversely  as  the  square  of  the  distance.  But  we  must  never 
forget,  in  our  investigations,  —  what  all  will  allow  to  be  an  axiom 
as  sound  as  any  which  has  been  adopted  as  such, — that  the 
velocity  of  a  falling  body  is  accelerated  from  the  beginning  of  the 
fall ;  that  is,  there  is  no  part  of  the  space  passed  over,  however 
small,  in  which  the  motion  is  not  accelerated  ;  and  for  this 
reason,  the  determination  will  become  absolute  and  conclu- 
sive, that  the  rate  or  intensity  of  gravity,  varies  inversely  as  the 
distance  varies.  But  these  principles  will  receive  farther  consid- 
eration and  elucidation  in  the  proper  place. 

Now  I  will  not  hesitate  to  declare  that  a  more  bungling,  erro- 
neous and  hap-hazard  exposition  of  the  law  of  gravity,  could 
scarcely  have  been  devised  by  the  ingenuity  of  man,  than  that 
put  forth  by  Maclaurin,  which  has  so  long  stood  the  test  of  time, 
for  surely  no  mental  or  physical  operation  has  affected  it.  And 


ON    THE    LAW    OF    GRAVITY.  139 

on  reading  it,  one  might  be  well  supposed  to  exclaim,  Is  it  no 
matter,  —  of  no  consequence,  upon  \v%at  premises  the  fabric  of 
astronomy  is  reared,  or  in  what  quagmire  its  foundations  are  laid  ? 
And  is  it  possible  that  such  an  exposition  could  have  thrown 
mankind  into  those  inexplicable  difficulties  under  which  they 
have  been  laboring  for  the  last  two  centuries  ? 

Mr.  Maclaurin,  after  setting  forth  in  his  premises  his  unity  of 
distance,  namely,  the  surface  of  the  earth  ;  his  unity  of  time, 
viz.,  one  second ;  his  unity  of  space,  viz.,  15TV  Parisian  feet ;  and 
the  distance  of  the  moon,  viz.,  60  units  of  distance ;  proceeds 
thus  :  "  And  her  (the  moon's)  gravity  increasing  in  proportion  as 
the  square  of  the  distance  from  the  centre  of  the  earth  decreases, 
her  gravity  would  be  60  X  60  (3600)  times  greater  near  the  sur- 
face of  the  earth  than  at  her  mean  distance,  and  therefore  would 
carry  her  through  60  X  60  X  15  iV  Parisian  feet,  (3600  units  of 
space,)  in  a  minute,  near  the  surface." 

Now  what  the  author  actually  meant  by  this  passage  must 
perhaps  rest  somewhat  upon  conjecture.  "Her  gravity  increas- 
ing inversely  as  the  square  of  the  distance  decreases,  her  gravity 
would  be  3600  times  greater  near  the  earth  than  at  her  mean 
distance."  If  he  intended  to  convey  the  idea  that  the  ponderos- 
ity of  the  moon  would  be  3600  times  greater  at  the  surface  of 
the  earth  than  at  the  distance  of  the  moon,  he  is  entirely  wel- 
come to  the  conjecture,  and  to  the  consequences  that  would 
follow,  were  the  conjecture  true  in  point  of  fact.  The  author 
surely  could  not  intend  that  a  body  commencing  its  fall  at  the 
distance  60,  would,  when  it  arrived  at  the  distance  1,  have  the 
same  velocity  that  a  body  would  then  have  that  commenced  its 
fall  at  the  distance  1 ;  for  it  is  presumed  that  he  defines  gravity 
after  the  manner  of  other  authors  ;  viz.,  that  it  is  the  force  which 
urges  the  body  towards  the  centre  of  gravity,  and  not  the  motion 
or  velocity  produced  by  gravity.  If  by  any  possible  construction 
we  could  suppose  that  the  author  would  only  infer  that  if  two 
bodies  commenced  their  fall  at  the  same  time,  one  at  the  surface 
of  the  earth,  and  the  other  at  the  distance  of  the  moon,  that  at 
any  given  instant  of  the  fall,  the  velocity  of  that  at  the  surface 
of  the  earth  would  be  3600  times  greater  than  that  at  the  distance 
of  the  moon,  it  would  be  strictly  true  in  point  of  fact,  but  his 
premises  assumed  do  not  permit  us  to  assign  such  as  his  mean- 
ing or  intention  ;  for  he  says,  "  her  gravity,  increasing  in  propor- 
tion as  the  square  of  the  distance  from  the  centre  of  the  earth 
decreases,  would  be  3600  times  greater  near  the  surface  of  the 
earth  than  at  her  mean  distance." 

And  it  is  only  upon  the  principle  that  the  rate  of  gravity  varies 
inversely  as  the  distance,  (not  inversely  as  the  square  of  the  dis- 


140  ON    THE    LAW    OF    GRAVITY. 

lance,)  that  the  velocity  of  a  body  at  the  distance  1,  is  3600 
times  greater  than  that  ar  the  distance  60,  at  any  given  instant  of 
their  fall,  both  commencing  at  the  same  time.  But  why  should 
that  be  assumed  as  a  fact  for  the  truth  of  which  we  are  seeking, 
and  the  truth  of  which  must  necessarily  be  the  result  of  the 
evidence  from  which  it  is  deduced  ?  Why  commence  by  say- 
ing that  the  gravity  of  the  moon  increases  in  proportion  as  the 
square  of  her  distance  decreases,  before  the  fact  is  ascertained, 
inasmuch  as  the  rate  of  gravity,  as  proportioned  to  distance,  is 
the  very  thing  we  wish  to  know  ?  But  the  premises  and  con- 
clusion intended  to  exhibit  cause  and  effect,  preceding  the  fol- 
lowing words,  viz.,  "  and  therefore  would  carry  her  through  60 
X  60  X  15T'2-  Parisian  feet,  (3600  unit  spaces,)  in  a  minute,  near 
the  surface  of  the  earth,"  would  be  equally  vague,  disjoined  and 
destitute  of  meaning,  even  though  it  were  a  fact,  that  the  rate  of 
gravity  increases  from  the  moon  to  the  earth  in  the  same 
proportion  as  the  square  of  the  distance  decreases ;  there  being 
no  premises,  conclusion  or  result  whatever  contained  in  the 
sentence, — nothing  except  absurdity.  And  it  appears  that  this 
part  of  the  exposition,  was  in  time,  suspected  of  discrepancy ; 
nevertheless,  the  law  must  not  fail;  and  hence  we  find,  in  the 
version  given  by  Mr.  Rees,  that  remarkable  passage  which  I 
have  already  quoted  and  given  over  to  the  world  for  investiga- 
tion arid  comment,  viz.,  "  but  this  force  (meaning  the  centripetal 
force  that  retains  the  moon  in  her  orbit)  being  known,  from  the 
elliptic  figure  of  her  orbit,  to  be  reciprocally  proportional  to  the 
square  of  the  distance,  would  impel  the  moon,  supposed  to  be 
at  the  surface  of  the  earth,  through  a  space  equal  to  60  X  60  X 
15TV  Paris  feet  (3600  unit  spaces)  in  one  minute."  This  version 
only  assumes  a  previous  knowledge  of  the  fact  that  the  rate  of 
gravity  is  inversely  as  the  square  of  the  distance;  viz.,  "it  being 
known  from  the  elliptic  figure  of  the  moon's  orbit." 

Now  we  know  that  by  the  law  of  falling  bodies,  a  body  fall- 
ing from  a  state  of  rest  at  the  surface  of  the  earth,  will  fall  3600 
times  as  far  in  one  minute,  as  it  will  in  the  first  second  of  time 
of  that  minute;  and  hence,  if  a  body  were  brought  from  the 
moon  and  were  to  commence  its  fall  near  the  surface  of  the 
earth,  it  would  doubtless  be  governed  by  the  law  of  falling 
bodies.  But  I  am  wholly  unable  to  discover  that  the  expositors 
have  in  any  way  or  manner  connected  this  fact  with  that  of  the 
Newtonian  law  of  gravity,  viz.,  that  the  rate  of  gravity  varies 
inversely  as  the  square  of  the  distance  varies. 

The  result  then  of  Mr.  Maclaurin's  reasoning  seems  to  be  this : 
that  a  body  falling  from  a  state  of  rest  near  the  surface  of  the 
earth  will  fall  15^  Parisian  feet  in  a  second  of  time ;  that  if 


ON    THE    LAW    OF    GRAVITY.  141 

the  moon  were  to  fall  from  a  state  of  rest  near  the  surface  of  the 
earth,  it  would  fall  3600  times  as  far  in  one  minute,  as  it  would 
in  one  second,  and  that  it  will  describe  3600  times  less  space  in 
one  second  than  in  one  minute,  and  therefore,  the  moon,  in  a 
second  of  time,  would  fall  by  her  gravity,  when  near  the  surface 
of  the  earth,  15T^  Parisian  feet ;  which  therefore  is  the  fact  with 
regard  to  the  gravity  of  terrestrial  bodies. 

Thus  far  I  have  thought  proper  to  comment  on  Mr.  Maclau- 
rin's  exposition  of  the  law  of  gravity ;  wishing  from  the  impor- 
tance of  the  subject,  (as  Mr.  Rees  justly  remarks,)  to  present  the 
same  in  as  bold  relief  as  possible,  with  a  view  to  solicit  an  in- 
vestigation of  the  subject.  And  whether  he  "  was  possessed  of 
many  whims  and  conceits,"  as  he  alleged  Kepler  to  have  been, 
when  comparing  him  with  the  illustrious  Newton,  must,  perhaps, 
be  inferred  from  his  works. 

Nevertheless,  I  think  he  has  presented  the  world  with  a  very 
strange  code  of  laws,  and  a  strange  body  of  philosophy ;  yet  no 
one  doubts  its  being  strictly  Newtonian.  It  is  not  consistent 
with  the  limits  of  Mr.  Rees?  Cyclopaedia  to  investigate  the  va- 
rious equations  that  have  been  explored  for  the  illustration  of 
these  laws  ;  and  in  fact,  to  go  into  an  elaborate  process  with  a 
view  to  present  such  as  were  adopted  by  Clairaut,  and  by  mathe- 
maticians in  general,  since  Clairaut  and  Euler  professed  their 
belief  in  the  Newtonian  laws,  would  be  an  onerous  task  indeed  ; 
for  already  has  the  rnoon  been  subject  to  between  thirty  and 
forty  equations  for  the  illustration  of  these  laws.  Mr.  Rees 
alleges,  in  his  preliminary  to  the  final  exposition  of  the  Newton- 
ian law  of  gravity  according  to  Maclaurin,  that  i4  is  well  known 
and  universally  allowed  (among  other  things)  that  the  efficacy 
of  the  force  of  gravity  increases  or  diminishes  in  the  inverse 
ratio  of  the  square  of  the  distance ;  upon  which  allegation  I  am 
bound  to  join  issue.  Mr.  Rees  also  makes  the  following  allega- 
tion, which  is  similar  in  its  consequences  to  the  foregoing,  and 
consequently  in  accordance  with  the  Newtonian  law  of  gravity, 
and  with  Mr.  Vince's  information  of  what  Sir  Isaac  Newton 
found,  viz. :  that  the  moon  is  drawn  by  its  centripetal  force  from 
a  tangent  to  its  orbit  towards  the  centre  of  the  earth  about  15^ 
Parisian  feet,  in  one  minute  of  time;  Or,  as  expressed  by  Mr. 
Vince  in  another  place,  "Sir  Isaac  Newton  found  that  if  the 
force  with  which  bodies  fall  upon  the  earth's  surface  were  ex- 
tended to  the  moon,  and  to  vary  inversely  as  the  square  of  the 
distance  from  the  centre  of  the  earth,  it  would  in  one  minute 
draw  the  moon  through  a  space  which  is  equal  to  the  versed 
sine  of  the  arc  which  the  moon  describes  in  one  minute.  He 
concluded,  therefore,  that  the  moon  was  retained  in  its  orbit  by 


142  ON    THE    LAW    OP   GRAVITY. 

the  same  force  as  that  by  which  bodies  are  attracted  upon  the 
earth."  Mr.  Rees  alleges  that  "  the  centripetal  force  of  the  moori 
revolving  in  its  orbit,  is  to  the  force  of  gravity  on  the  surface  of 
the  earth,  as  the  space  which  the  moon  would  describe  in  falling 
any  little  time,  by  her  centripetal  force  towards  the  earth,  if  she 
had  no  circular  motion  at  all,  is  to  the  space  which  a  body  near 
the  earth  would  describe  in  falling  by  its  gravity  towards  the 
same;" — which  sentence  perhaps  should  be  translated  thus: 
"  The  centripetal  force  of  the  moon  is  to  the  force  of  gravity  on 
the  surface  of  the  earth,  as  the  space  described  by  the  moon  in  a 
rectilinear  fall  towards  the  earth,  in  some  little  given  time,  caused 
by  the  moon's  centripetal  force,  is  to  the  space  described  in  an 
equal  time  by  a  body  falling  from  a  state  of  rest  near  the  surface 
of  the  earth  ; "  and  such  translation  appears  to  be  in  exact  ac- 
cordance with  what  Mr.  Vince  alleges  that  Sir  Isaac  Newton 
found.  And  in  corroboration  of  the  version  I  have  given,  Mr. 
Rees  proceeds  to  say,  that  "  the  space  at  the  moon  is  to  the  space 
at  the  surface  of  the  earth,  i.  e. —  the  centripetal  force  of  the  moon 
is  to  the  force  of  gravity  on  the  surface  of  the  earth,  —  as  the 
square  of  the  earth's  semi-diameter,  (square  of  the  distance  1,) 
is  to  the  square  of  the  semi-diameter  of  the  moon's  'orbit, 
(square  of  distance  60,)  which  is  the  same  ratio  as  that  of  the 
moon's  centripetal  force  to  the  same  near  the  surface  of  the 
earth."  And  hence  it  is  concluded,  and  that  too  from  such 
premises,  that  the  force  of  gravity  on  the  surface  of  the  earth, 
and  the  centripetal  force  of  the  rnoon,  are  identical ;  the  intensity 
varying  in  the  inverse  ratio  of  the  square  of  the  distance  as  we 
recede  from  the  centre  of  the  earth. 

If  then  it  has  been  ascertained  that  the  centripetal  force  acting 
upon  the  moon  would  (if  the  moon  had  no  circular  motion  in 
its  orbit)  draw  the  moon  from  a  state  of  rest  by  an  accelerated 
motion  towards  the  centre  of  the  earth  as  far  in  60  seconds  of 
time,  as  a  body  near  the  surface  of  the  earth  will  fall  from  a  state 
of  rest  in  one  second  of  time,  we  will  consent  that  it  furnishes 
indubitable  proof  of  the  identity  of  the  centrepetal  force  of  the 
moon  with  that  of  the  force  of  gravity  at  the  surface  of  the 
earth ;  the  intensity  or  efficiency  having  its  proportion  assigned 
in  the  powers  and  roots  of  numbers  in  some  given  inverse  ratio 
to  that  of  the  distance  from  the  centre  of  the  earth.  But  that 
such  fact  has  not  been  ascertained  or  disclosed  by  the  methods, 
or  in  the  manner  alleged  by  Mr.  Maclaurin,  Mr.  Rees  and  Mr. 
Vince,  will,  perhaps,  yet  become  too  manifest  for  contradiction. 
For  the  very  allegation  which  is  calculated  to  sustain  the  reason- 
ing upon  the  subject,  namely  :  that  gravity,  or  the  intensity  of 
gravity  varies  inversely  as  the  square  of  the  distance  varies, — 
throws  the  whole  subject  into  inexplicable  confusion,  admitting' 


ON    THE    LAW    OP    GRAVITY.  143 

of  no  rational  result  or  conclusion.  Nevertheless,  "  all  nature 
cries  aloud,  through  all  her  works,"  that  gravity  at  the  surface  of 
the  earth  which  causes  bodies  to  fall,  and  the  centripetal  force  of 
the  moon  which  retains  it  in  its  orbit,  are  identical,  or  one  and 
the  same  force,  varying  in  its  intensity  in  some  given  proportion 
to  the  inverse  of  the  distance,  which  ratio  is  assignable  in  the 
powers  and  roots  of  numbers.  Such  was  the  opinion  of  Kepler 
long  before  the  days  of  Newton,  when  he  declared  that  the  moon 
gravitated  towards  the  earth,  and  that  but  for  the  moon's  motion 
in  its  orbit,  the  earth  and  moon  would  come  together,  and  that 
too,  at  a  point  as  much  nearer  the  earth  than  the  moon,  as  the 
earth  is  greater  than  the  moon.  Such,  too,  was  the  opinion  of 
Kepler  when  he  disclosed  and  promulgated  those  eternal  laws, 
"  that  the  squares  of  the  periods  of  the  planets  are  as  the  cubes 
of  their  mean  distances  from  the  sun ;  and  that  the  motion  of 
any  given  or  individual  planet  varies  in  its  orbit  inversely  as  its 
distance  from  the  sun  varies."  And  this,  too,  may  be  indubita- 
bly proved  by  a  vast  amount  of  evidence. 

But  the  "  a  proof,"  which  Mr.  Vince  says  Sir  Isaac  Newton 
gave  Dr.  Halley,  to  me  is  no  proof  at  all ;  it  being  a  proof  in 
which  the  laws  of  Kepler  were  wholly  thrown  aside  and  disre- 
garded, and  in  which  the  law  of  falling  bodies  was  wholly 
misunderstood  and  perverted.  And  as  to  what  Mr.  Vince  says 
that  Sir  Isaac  Newton  found,  namely,  u  that  if  the  force  with 
which  bodies  fall  upon  the  earth's  surface  were  extended  to  the 
moon,  and  were  to  vary  inversely  as  the  square  of  the  distance  from 
the  centre  of  the  earth,  it  would  in  one  minute  draw  the  moon 
through  a  space  which  is  equal  to  the  versed  sine  of  the  arc 
which  the  moon  describes  in  one  minute."  I  shall,  in  some  fu- 
ture section,  —  in  which  the  element  of  deflection,  (namely,  the 
deflection  of  a  planet  from  a  tangent  to  its  orbit,)  which  has  been 
adopted  and  so  consequentially  considered  in  reference  to  central 
forces,  will  be  somewhat  liberally  treated  of,  —  endeavor  to  show 
that  Sir  Isaac  Newton  found  no  such  thing ;  for  the  reason  that 
no  such  thing  existed  in  nature. 

But  I  have  already  prolonged  this  section,  perhaps  quite  too 
far,  in  attempting  to  unravel  the  flimsy  woof  in  which  the  law 
of  gravity  has  been  enveloped,  or  in  cutting  the  meshes  in  which 
it  has  been  entangled,  inasmuch  as  the  shreds  and  remnants, 
however  carefully  saved,  can  be  of  no  future  use. 

But  what  has  Mr.  Rees,  or  any  other  commentator  on  the  law 
of  gravity,  done  with  the  point  to  be  mooted  ?  Where  are  their 
reasons,  their  evidence,  their  demonstrations,  that  show  what  the 
law  of  gravity  is  ?  Is  it  contained  in  a  subsequent  allegation  of 
Mr.  Rees,  "  that  this  force  is  known  from  the  elliptical  figure  of  the 


144  ON    THE    LAW    OF    GRAVITY. 

moon's  orbit,  to  be  reciprocally  proportional  to  the  square  of  the 
distance  ?  "  Or  is  it  contained  in  a  preceding  allegation,  which  is 
so  boldly  appended  to  certain  principles  which  are  well  known 
and  universally  allowed,  —  that  is,  is  it  proved  by  his  allegation, 
"  that  gravity  increases  as  the  square  of  the  distance  diminishes, 
and  that  it  diminishes  in  the  same  ratio  as  the  distance  increases  ?" 
Or  shall  we  look  for  the  proof  in  the  allegation  of  Mr.  Vince  and 
others,  that  "  Sir  Isaac  Newton  found,"  &c.  ?  And  if  we  do  not 
find  the  evidence  so  fully  developed  in  these  allegations  as  to 
make  it  highly  dangerous  to  doubt,  I  will  say,  let  us  reason  together 
upon  the  subject,  and  endeavor  to  comprehend  and  understand  the 
numerical  developments  of  the  great  physical  law  of  the  universe. 
For,  that  the  intensity  of  gravity,  is  proportioned  to  the  distance 
from  the  centre  of  gravity,  and  that  such  proportion  is  inverse,  or 
reciprocal  in  some  given  power  or  root  of  numbers,  as  that  of  the 
first  or  second,  or  some  other,  is  not  only  universally  allowed, 
but  is  a  necessary  consequence  synthetically  deduced  from  the 
great  Keplerean  laws,  as  well  as  from  all  phenomena  arising  from 
the  motion  of  the  heavenly  bodies.  What  this  proportion  be- 
tween distance  and  intensity  of  gravity  is,  Mr.  Rees  considers  of 
importance ;  at  least,  that  it  be  established  according  to  the  New- 
tonian hypothesis ;  and  hence  he  has  thought  proper  to  put  forth 
the  leading  facts  by  which  such  hypothesis  was  originally  explain- 
ed and  established.  But  if  others  should  think  it  of  equal  impor- 
tance that  the  law  of  gravity  should  be  determined  according  to 
the  truth,  I  have  no  fears  as  to  the  result. 

Much  stress,  to  be  sure,  has  been  placed  upon  the  fact  or  cir- 
cumstance, that  M.  Clairaut  found  out  a  very  ingenious  method 
of  extricating  himself  from  the  evils  of  a  long  rebellion  against 
the  Newtonian  law,  by  reconciling  himself  and  the  moon  to  its 
control ;  and  M.  Euler  escaped  by  retracting  his  opinion  in  def- 
erence to  the  judgment  of  M.  Clairaut ;  and  if  this  were  con- 
clusive evidence  in  respect  to  the  law,  surely  we  need  not  search 
farther,  and  may  rest  assured,  that  what  Mr.  Rees  calls  "  the 
various  equations  which  have  been  explored  for  the  illustration 
of  these  laws,"  are  in  reality  but  the  commencement  of  trouble. 

I  have  already  remarked  upon  Mr.  Rees'  premises  and  his 
supposed  consequent.  Nevertheless,  as  I  also  consider  the  sub- 
ject to  be  of  importance,  it  may  not  be  wholly  amiss  to  repeat 
the  same  and  append  a  comment  or  two,  by  way  of  investigation. 
The  truth  he  alleges  is  thus :  "  The  rectilinear  spaces  de- 
scribed in  any  given  time  by  a  falling  body  urged  by  any  powers, 
reckoning  from  the  beginning  of  its  descent,  are  proportionable 
to  those  powers."  And  his  supposed  consequent  (and  on  this 
point  the  investigation  is  particularly  required)  is  this :  "  Con- 


ON    THE    LAW    OF    GRAVITY.  145 

sequently  the  centripetal  force  of  the  moon,  revolving  in  its  orbit, 
is  to  the  force  of  gravity  on  the  surface  of  the  earth,  as  the  space 
which  the  moon  would  describe  in  falling,  any  little  time,  by  her 
centripetal  force  towards  the  earth,  if  she  had  no  circular  motion 
at  all,  is  to  the  space  which  a  body  near  the  earth  would  describe 
in  falling  by  its  gravity  towards  the  same." 

Now  although  the  author  has  confused  the  expressions  in 
which  his  supposed  consequent  is  couched,  beyond  any  of  his 
predecessors,  (for  what  reason  I  know  not,)  nevertheless,  it  may 
perhaps  be  rendered  so  as  to  obtain  whatever  of  meaning  it 
may  contain,  and  at  the  same  time  be  in  accordance  with  the 
views  of  the  illustrious  author  of  the  law  of  gravity,  namely,  that 
the  centripetal  force  exerted  on  the  moon  will  be  to  the  force 
of  gravity  at  the  surface  of  the  earth,  as  the  fall  of  the  moon 
from  a  tangent,  in  some  little  time,  (say  one  second,)  will  be  to 
the  space  described  by  a  body  falling  from  a  state  of  rest  at  the 
surface  of  the  earth  in  the  same  time.  Such  would  surely  be 
the  case  if  the  intensity  of  gravity  were  inversely  as  the  square 
of  the  distance.  But  the  question  in  respect  to  what  proportion 
the  intensity  of  gravity  bears  to  the  distance,  is  the  one  to  be 
solved,  and  that  too,  through  the  medium  of  the  phenomena 
which  gravity  produces.  The  law  of  gravity  then  is  not  to  be 
assumed,  a  priori,  but  must  aw.ait  its  development  in  and  through 
its  effects,  or  the  phenomena  which  it  causes,  if  so  be  it  may 
thus  be  obtained.  If  Mr.  Rees'  deduction  were  true,  it  would, 
as  he  observes,  be  obvious  that  the  centripetal  force  of  the  moon 
is  to  the  force  of  gravity  on  the  surface  of  the  earth  as  the  square 
of  the  earth's  semi-diameter  is  to  the  square  of  the  semi-diam- 
eter of  the  moon's  orbit ;  which  would  be  the  same  ratio  as  that 
of  the  moon's  centripetal  force  to  the  same  force  near  the  surface 
of  the  earth  ;  and  such  centripetal  force  would  in  such  case  be 
sufficiently  identified  with  that  of  gravity  near  the  surface  of  the 
earth,  as  was  fairly  suggested  by  Kepler.  But  I  think  Mr.  Rees' 
conclusion  of  this  part  of  his  exposition,  is  quite  too  vague  and 
full  of  unknown  consequents  to  merit  very  serious,  attention; 
and  perhaps  but  little  knowledge  can  be  extracted  from  the  same, 
except  it  be  of  the  confusion  of  the  subject  under  investigation. 
In  treating  of  the  identity  of  the  centripetal  force  of  the  moon, 
and  that  of  gravity  near  the  earth,  he  says,  "  these  two  forces 
consequently  are  not  different,  but  they  are  one  and  the  same  ; 
for  if  they  were  different,  bodies  acted  upon  by  the  two  forces 
conjointly,  would  fall  towards  the  earth  with  a  velocity  double  to 
that  arising  from  the  sole  power  of  gravity.  It  is  evident  there- 
fore, that  the  moon's  centripetal  force  by  which  she  is  retained 
in  her  orbit  and  prevented  from  running  off  in  tangents,  is  the 
19 


146  ON    THE    LAW    OF    GRAVITY. 

very  power  of  gravity  of  the  earth  extended  thither."  Now  per- 
haps there  are  those  who  can  discover  some  explanation  or  elu- 
cidation of  the  subject  in  the  last  quotation,  and  to  such  it  must 
be  referred ;  while  it  is  my  business  to  admire  the  unerring  pre- 
cision with  which  nothing  but  error  has  been  produced. 

Having  in  this  work  treated  somewhat  liberally  of  the  law  of 
falling  bodies,  more  with  a  view  of  disclosing  those  errors  which 
have  resulted  from  attempts  to  elicit  the  law  of  gravity  from  a 
consideration  of  the  phenomena  incident  to  falling  bodies,  than 
to  obtain  therefrom  the  law  of  gravity,  (believing  that  the  law  of 
gravity  may  be  much  more  easily,  simply,  and  satisfactorily  de- 
duced from  a  direct  consideration  of  the  Keplerean  laws,)  I 
shall  not  do  more  in  this  place  than  to  offer  a  few  brief  and  de- 
sultory hints  in  respect  to  the  law  of  falling  bodies.  I  will  use 
figure  1  to  denote  unity ;  what  I  will  call  one  moment  of  time 
may  be  considered  as  being  one  second  ;  the  moon  shall  be  con- 
sidered at  the  distance  1  from  the  centre  of  the  earth ;  and  con- 
sequently, the  surface  of  the  earth  will  be  at  l-60th  of  1  from 
the  centre  of  the  earth,  or  centre  of  attraction  ;  the  amount  of 
space  described  in  one  moment  by  a  body  falling  from  a  state  of 
rest  at  the  distance  of  the  moon,  or  distance  1,  will  be  called  one 
space ;  hence  the  amount  described  in  one  moment  by  a  body 
falling  from  a  state  of  rest  at  the  surface  of  the  earth,  or  at  the 
distance  l-60th  of  1,  will  be  3600  spaces,  for  the  number  of 
spaces  described  in  the  given  time  is  necessarily  reckoned  in 
the  sum  of  the  odd  numbers,  commencing  with  the  unit  space 
described  in  the  unit  of  time  by  the  body  whose  distance  is 
assumed  at  unity  or  1 ;  and  hence,  the  number  of  spaces  de- 
scribed in  the  unit  of  time  is  always  (properly  considered)  the 
inverse  of  the  square  of  the  distance,  (if  so  be  that  the  greater  dis- 
tance is  assumed  at  unity  •)  that  is,  making  the  body  whose  dis- 
tance is  assumed  at  unity  the  standard,  the  proportional  amount 
of  space  passed  over  in  the  given  unit  of  time,  will  always  be 
inversely  as  the  square  of  the  distance,  (but  not  of  the  time,)  and 
it  is  my  purpose  to  show  that  it  is  always  the  square  of  the 
whole  amount  of  force  applied  during  the  fall.  In  this  example, 
then,  the  body  at  the  distance  1,  or  distance  of  the  moon,  is  the 
standard ;  and  the  time  of  the  operation  is  one  moment,  or  one 
second  ;  and  in  that  time  the  amount  of  force  expended  on  the 
body  from  the  commencement  of  the  fall  is  1,  —  so,  also,  the  rate 
or  intensity  of  the  force  is  1,  the  body  at  the  surface  of  the  earth 
is  at  the  distance  of  l-60th  part  of  1,  and  the  time  of  the  opera- 
tion is  one  moment  or  one  second;  and  it  is  our  business  to  as- 
certain the  ratio  or  intensity  of  the  force  at  the  surface  of  the 
earth  compared  to  that  of  unity,  or  that  exerted  at  the  distance 


ON    THE    LAW    OF    GRAVITY.  147 

of  the  moon ;  as  also  the  proportional  amount  expended  on  the 
two  bodies  in  the  second  or  unit  of  time ;  that  expended  at  the 
distance  of  the  moon  being  1. 

It  having  been  well  proved  by  Galileo  and  others,  that  the 
numerical  law  of  falling  bodies  may  be  simply  presented  thus, 

11111     1 
1    3   5   7    9   11 

and  so  on,  ad  infinitum,  from  the  commencement  of  the  fall ;  the 
uniform  figure  1  denoting  an  equal  division  of  the  time,  and 
also  of  the  amount  of  force  expended  in  the  fall,  whether  the  in- 
tensity of  the  force  be  much  or  little  ;  which  time  or  force  thus  di- 
vided, may  be  called  the  equal  times  or  the  equal  forces;  and  fig- 
ure 1  in  the  series  of  odd  numbers  denoting  the  amount  of  space,  or 
the  space  described  by  the  body  in  the  first  time,  (called  the  unit  of 
time,)  or  while  operated  upon  by  the  first  force,  or  unit  of  space ; 
each  succeeding  odd  number  denoting  the  number  of  unit 
spaces  described  in  another  unit  of  time  or  unit  of  force  ;  and 
the  sum  of  all  the  odd  numbers  being  the  square  of  the  sum  of 
all  the  units  of  time,  or  of  all  the  units  of  force  expended  during 
the  fall ;  —  I  say,  such  having  been  determined  or  found  to  be  the 
numerical  law  of  falling  bodies,  perhaps  but  few  propositions 
present  themselves  to  the  mind  which  can  be  more  readily 
solved  than  the  following,  viz.  the  amount  of  force  applied  from 
the  commencement  of  the  fall  during  any  given  time,  or  while 
passing  over  any  given  space,  accumulates  with,  or  in  the  same 
ratio  or  proportion  as  the  square  root  of  the  whole  space  then 
described.  Hence  the  whole  amount  of  force  that  has  been  ap- 
plied at  any  given  instant  of  the  fall,  or  at  any  given  point  of 
space,  is  the  square  root  of  the  whole  amount  of  space  described. 
And  this  principle  in  respect  to  force  and  space  as  applied  to 
falling  bodies,  is  invariable  and  universal ;  whether  the  intensity 
of  the  force  be  great  or  little  ;  or  whether  the  body  commenced 
falling  at  the  distance  1,  or  at  the  distance  60,  or  at  the  distance 
l-60th  from  the  centre  of  gravity.  Thus,  if  the  distance  of  the 
moon  be  assumed  at  unity,  then  the  space  described  in  a  given 
moment  of  time,  (however  short,)  called  the  unit  of  time,  will  be 
the  unit  of  space  also ;  and  acceleration  will  have  been  the  con- 
sequence, even  in  this  unit  of  space,  however  small ;  and  a  body 
commencing  its  fall  at  the  surface  of  the  earth  or  distance  l-60lh, 
will  describe  3600  of  those  units  of  space  in  the  same  unit  of 
time.  Hence,  at  the  surface  of  the  earth  the  amount  of  space 
described  in  the  unit  of  time  is  inversely  or  reciprocally  as  the 
square  of  the  distance  ;  and  this  principle  is  universal,  and  holds 
good,  whatever  the  distance  may  be  which  is  assumed  at  unity. 


148  ON    THE    LAW    OF    GRAVITY. 

This  inverse  or  reciprocal  attitude  or  position  of  time  and  space, 
(the  two  passive  elements  observed  in  the  phenomena  and  used 
in  the  investigation  of  the  law  of  falling  bodies,)  is  necessarily 
presented  in  this  wise,  viz.  the  unit  of  time  being  the  same  at 
the  distance  of  the  moon  as  at  the  surface  of  the  earth,  if  we 
then  assume  the  unit  of  space  passed  over  in  such  unit  of  time 
to  be  at  the  distance  of  the  moon,  3600  of  them  will  be  passed 
over  in  the  unit  of  time  at  the  surface  of  the  earth.  But  if  we 
assume  that  as  the  unit  of  space  which  is  described  by  the  body 
at  the  surface  of  the  earth  in  the  unit  of  time,  then  the  amount 
of  space  described  at  the  distance  of  the  moon  in  the  unit  of 
time  will  be  less  than  the  unit  of  space  at  the  surface  of  the 
earth,  or  l-3600th  part  of  the  same.  Thus  the  amount  of  space 
described  in  a  given  time  or  unit  of  time  from  the  commence- 
ment of  the  fall  is  inversely  or  reciprocally  as  the  square  of  the 
distance  at  which  the  body  commences  its  fall.  Let,  then,  the 
time  be  given  (called  one  moment,  or  the  unit  of  time,  however 
long  or  short)  in  which  the  operation  is  to  be  performed.  Let 
the  space  to  be  described  in  such  moment  (from  the  commence- 
ment of  the  fall)  be  one  space  ;  and  when  thus  considered  in  the 
abstract,  the  original  form  of  the  mind  will  not  permit  us  to  con- 
sider either  the  rate  or  intensity  of  the  force,  or  the  whole 
amount  of  the  force  expended  while  describing  such  unit  of 
space,  to  be  other  than  unity,  or  more  properly  the  square  root 
of  unity  (for  we  are  endeavoring  to  ascertain  ratio  or  proportion 
as  developed  by  the  powers  and  roots  of  numbers.)  Then  sup- 
pose the  distance  at  which  the  fall  is  to  commence  to  be  less- 
ened to  l-60th  part,  and  consequently  that  the  number  of  units 
of  spaces  described  in  the  given  moment  will  be  increased  to 
3600,  made  up  of  the  sum  of  the  odd  numbers  in  their  order. 
Now  perhaps  no  proposition  can  become  more  manifest  in  its 
solution  than  that  the  amount  of  force  expended  in  the  given 
moment,  is  just  the  square  root  of  the  amount  of  the  space  de- 
scribed ;  for  the  force  being  uniform,  will  of  course,  in  order  to 
correspond  with  the  number  of  spaces,  be  divided  into  60  parts ; 
the  same  amount  being  expended  in  describing  the  three  spa- 
ces next  succeeding  the  first,  as  is  expended  in  the  first ;  and 
consequently  the  amount  of  force  expended  in  describing  the 
first  space  is  equal  to  that  expended  in  describing  an  equal 
space  or  the  unit  space  at  60  times  the  distance ;  and  conse- 
quently, in  either  case,  the  amount  of  force  expended  is  the 
square  root  of  the  amount  of  the  space  described ;  and  hence 
the  conclusion  is  inevitable,  that  the  rate  or  intensity  of  the  force 
is  inversely  as  the  distance.  And  now,  aside  from  demonstra- 
tion, whether  such  consideration  of  the  subject  be  more  or  less 


ON    THE    LAW    OF    GRAVITY.  149 

rational  or  satisfactory  to  the  mind,  than  that  which  has  so  long 
received  the  popular  sanction,  remains  to  be  determined,  viz. 
whether  physical  force  or  whether  passive  time,  which  has  no 
physical  agency  in  the  matter,  shall  have  the  credit  of  perform- 
ing the  operation,  viz.  that  of  urging  a  body  of  ponderous  mat- 
ter over  a  given  space  ;  for  the  fact  will  not  be  controverted  for 
a  moment  by  any  one,  that  by  the  popular  determination  in 
respect  to  the  law  of  falling  bodies,  time  is  a  far  more  efficient 
agent  than  force ;  and  that  if  one  body  commence  falling  at  the 
surface  of  the  earth,  and  another  at  the  distance  of  the  moon, 
when  they  shall  have  passed  over  equal  spaces,  59-60th  of  the 
fall  of  the  body  at  the  distance  of  the  moon  will  be  credited  to 
time  alone,  which,  in  fact,  has  only  been  passive,  awaiting  the 
operation,  without  having  any  physical  agency  whatever  in  the 
matter.  Nevertheless,  every  provident  person  rather  intends  to 
give  his  hireling  credit  for  the  efficient  force  expended  in  the 
accomplishment  of  an  operation,  than  for  the  time  that  may  hap- 
pen to  be  spent  while  the  operation  is  being  performed. 

Thus  have  I,  from  a  consideration  of  the  importance  of  the 
subject,  again  referred  to  the  law  of  falling  bodies,  and  its  con- 
sequent connection  with  the  law  of  gravity,  viz.  the  rate  or 
intensity  of  gravity  as  proportioned  to  the  distance  from  the  cen- 
tre of  gravity;  and  I  can  well  hope  that  the  importunity,  the 
urgency,  and  the  anxiety  which  I  have  displayed  on  the  subject, 
will  not  be  set  to  my  account  as  an  attempt  to  disparage  the 
judgment,  the  sagacity,  or  the  perception  of  an  enlightened 
world.  Whatever  "  Sir  Isaac  Newton  found,"  the  world  has 
been  extremely  loth  to  lose  ;  and  perhaps  it  might  be  rationally 
supposed,  that,  in  a  matter  so  simple,  Sir  Isaac  could  not  have 
found  amiss ;  and  hence,  that  the  subject  could  not  require  in- 
vestigation. 


SECTION    FIFTH. 


So  far  as  philosophy  depends  upon  numbers  (whether  nu- 
merically, mathematically  or  geometrically  considered)  for  its 
development,  it  is  in  vain  to  attempt  treating  of  subjects  with 
any  degree  of.  unity  of  purpose,  unless  we  assume  unity,  or  1,  as 
a  standard  in  all  our  investigations  on  which  to  rely  for  a  devel- 
opment of  the  truth  concerning  the  great  law  of  ratio  and  pro- 
portion between  quantities,  whether  of  the  same,  or  of  different 
denominations  ;  for  ratio  properly  flows  from  or  to  unity,  as  from 
or  to  ratio  in  unity;  and  in  unity  the  roots  and  powers  are 
equal.  Hence  if  unity  be  always  assumed  as  the  standard  in  the 
comparison  of  airquantities  susceptible  of  a  rational  compari- 


150 


ON    THE    LAW    OF    GRAVITY. 


son  by  means  of  numbers,  we  thereby  obtain  our  results  with 
ease  and  facility,  and  in  that  unity  of  purpose  by  which  they 
become  manifest  or  conclusive  to  the  mind.  This  I  have  en- 
deavored to  manifest  in  my  investigations  into  the  quadrature  of 
the  circle ;  as  also  when  rising 

"  Higher  in  the  line 
And  variation  of  determined  shape, 
Where  truth's  eternal  measures  mark  the  hound 
Of  circle,  cube  or  sphere." 

And  it  is  my  purpose  to  show  that  unity  is  the  proper  standard 
in  all  our  investigations  in  respect  to  the  laws  of  force  and  motion 
as  compared  with  those  passive  elements,  —  time  and  space ; 
whether  we  consider  the  laws  in  reference  to  impulsive  for- 
ces and  the  rate  of  motion  produced  thereby,  —  in  which  case  it 
has  been  fully  determined  and  shown  that  the  rate  of  motion 
or  velocity  produced  by  an  impulsive  force  is  the  root  of  the 
amount  offeree  applied,  —  as  that  four  times  the  impulsive  force 
gives  twice  the  rate  of  velocity ;  sixteen  times  the  impulsive 
force  gives  four  times  the  rate  of  velocity,  &c.  ; — or  whether 
we  consider  the  laws  of  force  and  motion  in  their  most  sim- 
ple and  common  occurrence  as  produced  by  human  or  finite 
operations;  viz.  in  directly  overcoming  the  ponderosity  of  a 
body  produced  by  the  force  of  gravity  in  raising  it  through  a 
given  space  by  a  uniform  motion,  (neither  accelerated  nor  retard- 
ed,) by  simple  human  force  or  power,  unaided  by  any  of  the  me- 
chanical powers,  properly  so  called,  as  that  of  the  lever,  or  wheel 
and  axle,  &c.,  in  which  case,  each  element  proper  to  be  consid- 
ered is  assumed  at  unity,  or  1,  as  that  of  the  ponderosity  of  the 
body,  —  the  given  space  passed  over,  —  the  time  of  the  operation, 
—  the  rate  or  intensity  of  the  force  employed,  —  the  rate  of  ve- 
locity with  which  the  body  moves,  —  as  also  the  whole  amount 
of  force  expended  in  the  whole  time  of  the  operation ;  —  or  wheth- 
er we  consider  the  application  of  the  mechanical  powers  in  rais- 
ing such  body  over  a  like  space  by  a  constant  and  uniform  mo- 
tion, but  by  the  constant  application  of  a  less  rate  or  intensity  of 
force;  in  which  case  the  time  required  will  be  inversely  or  recip- 
rocally as  the  rate  or  intensity  of  the  constant  force  applied ; 
thus,  if  the  rate  of  force  be  half  of  unity,  the  time  required  will 
be  twice  unity;  if  the  time  be  60,  the  rate  of  force  will  be  l-60th 
of  unity,  &c.  and  the  rate  of  velocity  will  be  as  the  rate  of  force. 
Thus  the  elements  of  ponderosity,  of  space,  and  of  the  whole 
amount  of  force  employed  in  the  operation,  will  be  as  those  in 
the  foregoing  example,  viz.  each  will  be  assumed  at  unity, 
while  those  of  time  and  the  rate  or  intensity  of  the  force  em- 
ployed will  be  the  inverse  ratio  or  reciprocal  of  each  other. 
Thus  the  great  lesson  to  be  derived  from  the  foregoing  is,  that 


ON    THE    LAW    OF    GRAVITY.  151 

the  same  amount  of  force  is  required  in  raising  the  body  over 
the  given  space,  whether  the  time  be  longer  or  shorter ;  or 
whether  it  be  performed  by  the  aid  of  the  mechanical  powers, 
properly  so  called,  or  not.  Time,  in  waiting  for  the  accomplish- 
ment of  the  object,  necessarily  waits  until  the  amount  of  force  or 
labor  required  shall  have  been  expended  in  the  accomplishment. 
Hence  the  whole  amount  of  force  expended  in  raising  the  body 
over  the  given  space,  will  be  the  same,  whether  the  time  be 
longer  or  shorter,  or  whether  the  rate  or  intensity  of  the  force  be 
greater  or  less.  If  we  attempt  to  investigate  the  law  of  falling 
bodies  towards  the  earth,  and  to  deduce  therefrom  the  law  by 
which  the  force  of  gravity  varies  in  proportion  to  distance  from 
the  centre  of  gravity  ;  or  if  we  attempt  to  investigate  the  laws  of 
force  and  motion  as  manifested  in  the  revolutions  of  the  heav- 
enly bodies,  and  proportion  the  elements  of  force  and  motion  to 
those  of  time  and  the  space  passed  over,  our  labors  will  scarcely 
receive  a  liberal  compensation  unless  we  adopt  unity  as  the 
standard  of  comparison.  In  an  investigation  of  the  law  of  fall- 
ing bodies  towards  the  earth,  with  a  view  to  ascertain  by  what 
ratio  the  force  of  gravity  varies  with  the  varying  distance  from 
the  centre  of  the  earth  or  centre  of  gravity,  some  of  the  premises 
on  which  to  proceed  with  the  investigation,  have  been  very 
properly  assumed ;  as  that  of  the  comparison  of  two  falling 
bodies  towards  the  earth, —  one  commencing  its  fall  from  a  state 
of  rest,  at  or  near  the  surface  of  the  earth,  and  the  other  at  some 
given  number  of  semi-diameters  of  the  earth  from  the  centre  of 
the  earth ;  as  that  of  four  times  the  semi-diameter  of  the  earth 
from  the  centre,  or  sixty  times  the  semi-diameter  of  the  earth 
from  the  centre,  being  the  mean  distance  of  the  moon,  &c.  And 
as  the  comparison  of  two  bodies  commencing  their  fall  at  dif- 
ferent distances  from  the  centre  of  gravity,  is  in  most  cases  suf- 
ficient for  all  the  essential  developments  required,  they  may  be 
denoted  by  the  letters  A  and  D ;  A  commencing  its  fall  at  or 
near  the  surface  of  the  earth,  which  distance  from  the  centre  of 
the  earth  or  centre  of  gravity,  is  called  unity  or  1,  —  while  D 
shall  commence  its  fall  from  several  times  the  distance  from  the 
centre  that  A  does,  as  at  four  times,  eight  times,  &c.,  or  at  sixty 
times,  which  is  the  distance  of  the  moon,  &c.,  both  commencing 
their  fall  at  the  same  instant  of  time ;  and  the  operation  to  be 
performed  and  completed  is  for  each  to  pass  over  an  equal 
space,  called  the  given  space,  whatever  difference  there  may  be 
in  the  time  of  the  passage  of  the  respective  bodies  over  this  given 
space. 

The  given  space  is  some  abstract  numerical  quantity,  denoted 
by  unity  or  1,  and  may,  if  any  prefer  so  to  consider  it,  be  de- 
noted by  some  arbitrary  measure,  as  one  rod,  that  being  very 


152  ON    THE    LAW    OF    GRAVITY. 

nearly  the  space  over  which  a  body  will  fall  from  a  state  of  rest 
at  or  near  the.  surface  of  the  earth,  in  the  first  second  of  time 
after  the  commencement  of  its  fall ;  or  in  which  a  body  at  the 
distance  of  the  moon  will  fall  in  the  first  sixty  seconds  of  time 
after  the  commencement  of  its  fall.  Such  given  space,  however, 
should  be  some  small  quantity  in  which  the  intensity  of  the 
force  applied  to  either  A  or  D,  during  the  fall  over  the  given 
space,  shall  not  be  increased  ;  or  so  that  there  shall  be  no  in- 
crease to  affect  our  calculations  and  the  results  arrived  at,  or 
make  them  different  from  a  perfect,  uniform  intensity,  for  so  it 
will  be  considered. 

It  will  be  perceived  that  the  object  or  final  result  of  our  re- 
searches is  to  ascertain  the  ratio  between  the  intensity  of  the 
force  of  gravity  applied  to  A,  and  that  which  operates  on  D,  dur- 
ing their  fall  over  the  given  space.  The  given  space  to  be 
passed  over  by  each  body,  is  an  element  in  common  between 
the  bodies ;  and  the  operation  or  labor  to  be  performed  by  the 
force  of  gravity,  is  to  pass  each  body  over  the  given  space,  or  an 
equal  space ;  and  as  the  time  of  the  passage  will  not  be  the 
same  in  respect  to  both  bodies,  we  wish  to  ascertain  what  rela- 
tion time  has  to  the  intensity  of  the  force  which  urges  the  bodies 
over  the  given  space  ;  what  relation  the  intensity  of  the  force 
has  to  the  acceleration  of  the  bodies  while  passing  over  the 
given  space,  or  to  the  instant  velocity  of  the  bodies  at  any  given 
instant  of  the  fall,  or  at  any  given  point  of  the  space ;  and 
hence,  what  is  the  ratio  of  the  intensity  of  the  force  of  gravity 
at  the  distance  of  D  from  the  centre  of  gravity,  to  that  at  the  dis- 
tance A ;  and  hence,  to  ascertain  the  ratio  or  proportion  which 
the  whole  amount  of  force  expended  upon  D  while  passing  over 
the  given  space,  bears  to  the  whole  amount  of  force  expended 
on  A  while  passing  over  the  given  space.  The  premises  upon 
which  Sir  Isaac  Newton,  Maclaurin,  and  other  mathematicians, 
proceeded  in  their  attempts  to  investigate  the  law  of  gravity,  viz. 
to  ascertain  the  intensity  of  the  force  of  gravity  as  proportioned 
to  the  distance  from  the  centre  of  gravity,  —  are  based  upon  the 
very  proposition  that  a  body  commencing  its  fall  at  or  near  the 
surface  of  the  earth  may  be  assumed  at  the  distance  of  unity  or 
1  from  the  centre  of  gravity,  viz.  from  the  centre  of  the  earth ; 
and  as  the  moon  is  sixty  semi-diameters  of  the  earth  from  the 
centre  of  the  earth,  hence  a  body  commencing  its  fall  at  the  dis- 
tance of  the  moon,  commences  at  the  distance  60  from  the  cen- 
tre of  gravity ;  and  a  body  at  the  surface  of  the  earth  will 
fall  from  a  state  of  rest  as  far  in  the  first  second  or  moment  of 
time,  as  the  moon  will  fall  from  a  tangent  to  its  orbit  in  60  sec- 
onds. These  proportional  distances  of  A  and  D  are  not  objec- 


ON    THE    LAW    OF    GRAVITY.  153 

tionable  otherwise  than  in  the  extension  of  a  numerical  formula 
adapted  to  so  high  a  number  as  60,  intended  to  aid  the  investi- 
gation by  an  exhibition  of  the  numerical  law  of  falling  bodies 
in  its  most  simple  form,  and  in  ringing  those  changes  upon  it 
best  calculated  to  place  the  phenomena  in  the  most  perspicu- 
ous attitude. 

I  shall  hence  present  the  most  simple  numerical  formula 
which  I  can  conceive  of,  in  which  the  time  of  the  fall  of  D  over 
the  given  space  will  be  eight  fold  that  of  A ;  or  in  which  the 
distance  from  the  centre  of  gravity  at  which  t>  commences  its 
fall,  is  eight  times  that  at  which  A  commences  its  fall.  This 
will  require  that  the  given  space  over  which  D  is  to  fall  shall  be 
divided  into  64  equal  parts ;  and  that  the  time  of  the  fall  of  D 
shall  be  divided  into  8  equal  parts,  called  moments,  each  mo- 
ment being  of  an  equal  length  to  that  employed  by  A  in  falling 
over  the  entire  space.  But  it  will  be  readily  seen  that  the  form- 
ula given  may  be  extended  so  as  to  denote  D  at  60  times  the 
distance  from  the  centre  of  gravity  that  A  is,  viz.  until  the  series 
denoted  by  the  constant  figure  1  shall  amount  to  60  places ; 
thereby  not  only  denoting  the  proportional  distance  of  D  to  that 
of  A  from  the  centre  of  gravity,  but  also  the  proportional  time 
of  the  fall  of  D  over  the  given  space  to  that  of  A,  (for  it  will 
readily  be  seen  that  the  proportional  distance  from  the  centre  of 
gravity  will  always  be  as  the  proportional  times  of  the  fall  over 
the  given  space.)  And  if  the  proportional  distance  or  time  of 
D  to  that  of  A  be  as  60  to  1,  the  given  space  over  which  D 
shall  pass  will  properly  be  divided  into  3600  equal  parts,  (or 
into  the  square  of  the  distance  or  of  the  time.)  And  hence  in  the 
last  moment  of  the  fall  of  D,  it  will  pass  over  59  of  the  equal 
parts  of  the  given  space.  The  formula,  then,  as  applied  to  D, 
may  be  as  follows  : 

11111  1  1  1 

1   2  3   4  5   6   7   8    9    10    11   12   13    14    15 

Thus  denoting  the  time  or  distance  of  D  to  be  eight  fold  that 
of  A ;  and  that  the  given  space  passed  over  by  D  is  divided  into 
64  equal  parts,  the  first  part  being  passed  over  in  the  first  mo- 
ment, three  equal  parts  in  the  second  moment,  five  in  the  third 
moment,  or  in  the  time  in  which  A  would  pass  over  the  whole 
64  equal  parts.  Now  the  number  of  equal  spaces  of  the  given 
space  passed  over  by  D,  is  as  well  the  square  of  the  distance  of 
D  as  of  the  time;  for  the  distance  is  as  the  time,  each  being  eight 
times  that  of  A. 

The  formula  also  presents  this  phenomenon :  viz.  at  the  end 
of  the  first  moment  figure  2  will  denote  the  rate  of  velocity  ; 
20 


154  ON    THE    LAW    OF    GRAVITY. 

figure  4  at  the  end  of  the  second  moment,  figure  6  at  the  end 
of  the  third  moment,  and  so  on  by  the  even  numbers.  Hence, 
at  the  end  of  the  first  moment  the  velocity  would  carry  the  body 
over  two  of  the  equal  spaces  in  one  moment,  by  a  uniform  mo- 
tion, without  the  aid  of  further  force  applied  ;  at  the  end  of  the 
second  moment  the  velocity  alone,  by  a  uniform  motion,  would 
carry  the  body  over  four  equal  spaces  in  one  moment,  without 
further  force  applied,  and  so  on.  If,  then,  the  whole  amount 
of  force  applied  to  D,  while  passing  over  the  given  space,  be 
divided  into  as  many  equal  parts  as  the  time  is  into  equal  mo- 
ments, so  that  the  several  equal  parts  shall  be  denoted  by  the  con- 
stant figure  1,  then  the  velocity  at  the  end  of  any  given  moment 
of  the  fall  may  be  said  to  be  twice  the  amount  of  the  whole 
force  expended  from  the  commencement  of  the  fall.  If,  then, 
the  whole  amount  of  force  be  thus  divided  into  equal  parts,  we 
find  that  the  whole  number  of  those  parts  is  the  square  root  of 
the  whole  number  of  spaces  into  which  the  given  space  is  di- 
vided ;  and  hence  that  an  equal  amount  of  force  is  expended  in 
such  equal  amount  of  time.  Calling,  then,  the  equal  amounts 
of  force  in  the  equal  moments  of  time,  the  forces  expended  dur- 
ing the  fall  over  the  given  space,  the  sum  of  all  the  equal  spaces 
into  which  the  given  space  is  divided  is  the  square  of  the  sum 
of  the  equal  forces.  Hence,  the  whole  amount  of  space  passed 
over  at  any  given  instant  of  time  or  at  any  given  point  of  space 
during  the  fall,  is  the  square  of  the  whole  amount  of  force  that 
has  then  been  expended  during  the  fall ;  and  hence  the  whole 
amount  of  the  given  space  is  ihe  square  of  the  whole  amount  of 
force  expended  while  the  body  is  passing  over  the  given  space ; 
and  this,  too,  whether  the  time  be  longer  or  shorter.  Thus,  pas- 
sive time  only  waits  for  force  to  accomplish  the  labor,  or  object 
to  be  accomplished ;  and  in  such  amount  of  time  as  shall  be 
required  according  to  the  intensity  of  the  force  employed ;  and 
that  loo  without  the  slightest  interference  on  the  part  of  time, — 
the  whole  amount  of  the  time  of  the  operation  only  varying 
from  unity  in  the  reciprocal  or  inverse  ratio  to  the  intensity  of 
the  force,  precisely  in  accordance  with  an  operation  by  the  me- 
chanical powers  (as  of  the  wheel  and  axle)  in  raising  a  ponder- 
ous body. 

Such,  then,  is  the  manifest  operation  of  a  falling  body,  which 
seems  to  be  much  more  in  accordance  with  all  we  know  of  the 
laws  of  force  and  motion,  whether  mathematically  or  experiment- 
ally obtained,  than  that  hypothesis  which  has  been  so  carelessly 
guessed  off  as  the  physical  basis  of  astronomy,  in  which  just 
fifty-nine  sixtieths  of  the  space  passed  over  by  a  body  commenc- 
ing its  fall  at  the  distance  of  the  moon,  or  distance  60,  is  taken 


ON    THE    LAW    OF    GRAVITY.  155 

or  withheld  from  the  labor  of  active  or  effective  force,  and  is 
credited  to  passive  time,  —  thus  making  time  far  more  active 
and  effective  than  force,  and  far  the  greater  operative.  The 
given  space,  whether  applied  to  A  or  D,  is  unity,  and  is  the 
square  of  the  whole  amount  of  force  expended  during  the  fall 
either  of  A  or  D  over  the  given  space;  and  in  unity,  the  roots 
and  powers  are  equal ;  hence  the  amount  of  force  applied  to  D 
is  equal  to  that  applied  to  A,  while  passing  over  the  given  space, 
A  being  the  standard,  all  its  elements  being  assumed  at  unity  ; 
hence,  the  rate  or  intensity  of  the  force  applied  to  A  is  as  the 
whole  time  of  the  fall  of  A,  (or  figuratively,  by  the  adoption  of 
ratio  in  unity,  the  intensity  of  the  force  applied  to  A  may  be 
said  to  be  in  the  inverse  ratio  or  in  the  reciprocal  ratio  of  the 
whole  time  of  A's  fall.)  And  so  in  respect  to  D,  —  for  as 
the  intensity  of  the  force  varies  from  unity,  it  varies  in  the  in- 
verse ratio,  or  in  the  reciprocal  ratio  to  that  of  the  whole  time ; 
for  such  is  the  universal  law  or  condition  in  respect  to  force  and 
motion  in  time,  or  as  applicable  to  time. 

Here,  then,  is  a  principle  which  always  holds  good  in  regard 
to  the  effect  of  the  force,  (viz.  the  amount  of  labor  performed,) 
whatever  be  the  intensity  of  the  force  employed,  —  whether  it 
be  greater  or  less  than  unity,  —  for  if  the  farther  body  be  assumed 
at  unity,  the  intensity  of  the  force  applied  to  the  nearer  will  be 
more  than  unity  ;  and  the  time  of  its  fall,  less  than  unity.  The 
principle  is  this :  in  a  falling  body  in  which  the  velocity  is  con- 
stantly accelerated  from  the  commencement  of  the  fall,  the 
amount  of  space  passed  over  at  any  given  inslant  of  time,  or  at 
any  given  point  of  space,  is  the  square  of  the  whole  amount  of 
force  which  has  been  expended  on  the  body  from  the  com- 
mencement of  the  fall.  And  this  principle  holds  good  whether 
the  body  commence  its  fall  at  the  distance  1,  or  at  the  distance 
60,  or  at  any  other  given  numeral  distance  from  the  centre  of 
gravity.  Were  it  otherwise,  the  whole  amount  of  force  applied  to 
a  body  while  passing  over  a  given  space  from  the  commencement 
of  its  fall,  would  always  be  in  the  inverse  ratio  to  the  distance 
from  the  centre  of  gravity.  If  the  distance  were  8,  the  amount 
of  force  required  to  pass  the  body  over  a  given  space  from  a 
state  of  rest,  would  be  but  l-8th  of  what  would  be  required 
at  the  distance  1  ;  so  at  the  distance  60,  the  amount  of  force  re- 
quired would  be  but  l-60th  of  what  would  be  required  at  the 
distance  1,  &c.  But  such  is  the  deduction  of  Sir  Isaac  Newton 
and  his  followers  in  respect  to  the  law  of  falling  bodies,  and 
consequently  in  respect  to  the  law  of  gravity,  viz.  the  intensity 
of  gravity  as  proportioned  to  the  distance  from  the  centre  of 
gravity.  But  it  is  an  operation,  an  effect  produced  by  force, 


156  ON    THE    LAW    OF    GRAVITY. 

for  which  we  are  seeking,  with  a  view  to  ascertain  or  deduce 
its  proportional  intensity  at  different  distances  from  the  centre  of 
gravity  ;  and  although  we  may  figuratively  speak  of  the  opera- 
tion of  time,  of  the  effect  of  time,  &c.  (not  of  the  intensity,)  nev- 
ertheless, such  figures  are  not  properly  used  or  adopted  in  a 
mathematical  investigation  of  those  operations  and  effects  which 
take  place  in  time ;  and  it  would  seem  far  more  natural  in  an 
investigation  of  the  intensity  of  gravity,  that  the  space  passed 
over  at  any  given  instant  of  time,  or  at  any  given  point  of  the 
space  from  the  commencement  of  the  fall,  should  he  referred  to 
the  active  or  operative  element  of  force,  rather  than  to  the  pas- 
sive element  of  time  ;  that  the  space  passed  over  should  be  call- 
ed the  square  of  the  whole  force  expended  from  the  commence- 
ment of  the  fall,  which  is  true  in  each  individual  ease,  (what- 
ever the  time  may  be,)  rather  than  that  the  space  passed  over 
should  be  called  the  square  of  the  time  which  has  elapsed  dur- 
ing the  fall,  —  which  can  never  be  true,  except  the  time  be  assum- 
ed at  unity.  For  at  the  distance  1,  if  the  given  space  be  the 
square  of  the  whole  time,  at  the  distance  60,  the  given  space 
will  be  the  square  of  only  1-60-th  of  the  whole  time.  Hence,  if 
the  time  required  at  the  distance  lr  for  the  body  to  fall  over  the 
given  space,  be  increased  sixty  fold  or  to  60  times,  in  conse- 
quence of  a  body's  commencing  its  fall  at  60  times  the  dis- 
tance, then  to  be  sure,  in  order  that  the  number  of  spaces  shall 
be  the  square  of  the  number  of  times,  the  given  space  at  the 
distance  60  must  be  divided  into  3600  equal  spaces ;  neverthe- 
less, a  formula  so  constructed,  in  which  the  space  to  be  passed 
over  shall  only  be  referred  to  the  time  of  the  passage,  affords  of 
itself  alone  no  direct  clue  to  the  proportional  intensity  of  the 
force  of  gravity  at  different  distances  from  the  centre  of  gravity. 

Let  us  then  always  recollect,  or  keep  in  mind,  the  all-im- 
portant principle,  so  full  of  meaning  and  so  directly  material 
to  the  issue,  and  which  will  disclose  itself  in  an  infinity  of  the 
most  happy  results  and  consequences,  viz.  that  the  space  pass- 
ed over  at  any  given  instant  of  time,  or  at  any  given  point  of 
space  from  the  commencement  of  the  fall,  is  the  square  of  the 
whole  amount  of  force  which  has  then  been  expended  during 
the  fall ;  or  in  a  falling  body  in  which  the  velocity  is  continu- 
ally accelerated  by  the  force  of  gravity,  the  whole  space  passed 
over  is  the  square  of  the  whole  amount  of  force  that  has  been 
expended  during  the  fall ;  nor  should  we  forget  the  important 
principle  that  the  rate  or  intensity  of  the  force  applied  is  in  the 
inverse  ratio,  or  in  the  reciprocal  ratio  to  the  whole  amount  of 
time  in  which  the  force  is  applied  ;  nor  the  no  less  important 
principle,  that  the  time  is  as  the  distance ;  all  which  principles 


ON    THE    LAW    OF    GRAVITY.  157 

are  readily  solved  upon  the  simplest  of  the  mechanical  powers, 
as  that  of  the  lever,  or  wheel  and  axle  ;  nor  do  we  find  that  the 
immutable  law  based  upon  those  simple  principles,  has  in  any 
case  been  abrogated  or  altered.  And  if  these  simple  principles 
were  once  applied  to  the  mechanism  of  the  heavens,  in  lieu  of 
that  strange  complication  of  human  contrivance  which  now 
assumes  the  control,  new  beauties  would  be  displayed,  not  only 
for  the  wonder  and  admiration  of  the  learned,  but  for  a  vast  mul- 
titude who  have  been  wholly  unable  to  discover  much  beauty 
in  the  works  of  creation  through  the  medium  presented. 

Upon  the  foregoing  simple  principles  we  find  this  result ;  that 
if  one  body  commence  falling  at  the  distance  1,  at  the  same 
time  that  a  body  commences  falling  at  the  distance  60,  the 
velocity  of  the  body  at  the  distance  1  will  be  3600  times  as 
great  as  that  at  the  distance  60,  at  any  given  instant  of  time 
during  their  continued  fall.  For  the  force,  viz.  the  rate  or  in- 
tensity of  gravity,  at  the  distance  60,  being  only  l-60th,  or, 
.01666,  will  be  .0002777,  or  the  l-3600th  part  of  the  given  space, 
which  is  the  square  of  l-60th  part  of  the  whole  force  applied  to 
A  while  passing  over  the  given  space  ;  the  final  result  of  which 
will  be,  that  an  equal  amount  of  force  will  be  expended  on  D 
while  passing  over  the  given  space,  that  will  be  expended  on  A 
while  passing  over  the  given  space.  Nor  can  the  operation  be 
carried  on  otherwise,  being  established  in  the  immutable  fitness 
of  things.  Spite  of  all  this,  the  Newtonian  method  of  investiga- 
tion makes  the  intensity  of  the  force  the  inverse  ratio  of  the 
square  of  the  reciprocal  of  the  time,  as  well  as  of  the  square  of 
the  reciprocal  of  the  distance;  all  of  which  is  equally  absurd; 
and  hence,  he  allows  but  l-60th  part  of  the  force  at  the  distance 
60,  that  is  allowed  at  the  distance  1,  to  perform  the  same  opera- 
tion or  same  quantity  of  work.  Thus,  if  one  of  the  simplest 
and  best  established  facts  in  creation  is  to  be  regarded,  viz.  that 
it  is  force  which  moves  ponderous  bodies,  and  that  it  requires 
a  like  amount  of  effective  force  to  move  a  given  body  over  a 
given  space,  whether  it  be  in  a  longer  or  shorter  time,  if  the 
time  be  only  passive,  the  whole  time  standing  as  the  reciprocal 
to  the  rate  or  intensity  of  the  force  applied,  waiting  its  opera- 
tion ;  then  at  the  distance  of  the  moon  or  distance  60  from  the 
centre  of  gravity,  the  intensity  of  the  force  of  gravity  is  J-60th 
as  great  as  at  the  surface  of  the  earth,  or  distance  1  from  the 
centre  of  gravity.  Thus  the  proportional  velocities  at  any  given 
instant  of  time  of  two  bodies  commencing  their  fall  at  the  same 
time,  but  at  different  distances  from  the  centre  of  gravity,  are  as 
the  spaces  generated,  or  rather,  as  the  space  generated  by  each 
respectively ;  and  the  whole  space  generated  by  either  at  any 


158  ON    THE    LAW    OF    GRAVITY. 

given  instant  of  time,  or  point  of  space,  is  the  square  of  the 
whole  amount  of  force  that  shall  have  been  expended  in  urging 
the  fall  thus  far.  If,  then,  the  foregoing  is  a  true  exposition  of 
the  law  of  falling  bodies,  it  is  also  at  the  same  time  a  very 
easy  and  simple  exposition  of  the  law  of  gravity,  determining 
that  the  rate  or  intensity  of  gravity  varies  inversely  as  the  dis- 
tance varies  ;  and  as  it  is  demonstrable  that  such  law  of  gravity 
would  afford  to  the  motion  of  the  moon's  apogee  just  four  times 
the  force  afforded  by  the  Newtonian  law,  (and  consequently 
twice  the  motion,)  the  adaptation  of  such  law  would  have  saved 
Clairaut  much  perplexity  in  creating  factitious  force  in  ac- 
counting for  the  observed  motion  ;  and  it  would  have  deprived 
others  of  the  privilege  of  creating  factitious  force  whenever  they 
found  a  revolving  body  not  to  quadrate  with  the  Newtonian 
law.  The  Newtonian  method,  then,  is  erroneous.  But  many 
conspicuous  names  are  engraved  thereon,  at  an  awful  height ; 
but  those  of  Galileo  and  Kepler  are  not  there  ;  and  whoever 
will  search  for  those  names  must  wander  upon  the  rock  far  in 
the  back-ground. 

But  the  subject  is  vastly  important ;  and  it  will  be  my  en- 
deavor to  throw  sufficient  evidence  before  the  public  to  induce  a 
thorough  investigation  of  the  subject,  even  though  I  should  be 
unjustly  accused  of  resorting  to  the  ingenious  method  of  the 
painter  for  distinguishing  the  man  from  the  bear,  which  he  had 
painted  on  the  same  canvass.  I  would  to  Heaven  that  the  world 
would  investigate  the  Newtonian  method  of  deducing  the  law 
of  gravity ;  for  if  the  world  shall  still  prefer  that  the  deduction 
shall  be  drawn  from  a  consideration  of  the  law  of  falling  bodies 
towards  the  earth,  in  lieu  of  being  drawn  from  a  far  more  nat- 
ural and  simple  method,  it  is  certainly  proper  that  we  should  take 
heed  to  our  steps,  even  while  travelling  in  the  road  which  we 
may  prefer. 

I  need  not  here  refer  to  the  well  known  laws  of  force  and  motion 
as  applicable  to  the  mechanical  powers,  as  that  of  the  wheel  and 
axle,  the  windlass,  &c.,  in  which  the  time  is  as  the  distance,  or 
length  of  lever,  where  the  force  is  applied,  and  in  which  the  in- 
tensity of  the  force  applied  is  inversely  as  the  time;  and  conse- 
quently inversely  as  the  distance.  And  it  would  seem,  also,  a 
matter  of  supererogation,  to  refer  to  the  fact  that  the  same  law 
prevails  in  respect  to  bodies  falling  towards  the  earth ;  inasmuch 
as  it  is  so  directly  manifest  in  the  phenomena  of  falling  bodies, 
as  presented  to  view  by  the  numerical  formula  by  which  the  law 
is  so  admirably  explained;  and  more  especially,  as  it  is  fully  as- 
sented to  in  that  very  process  by  which  it  has  been  supposed  the 


ON    THE    LAW    OF    GRAVITY.  159 

fact  has  been  deduced,  that  the  force  of  gravity  is  inversely  as  the 
square  of  the  distance  ;  that  is,  that  a  body  at  the  distance  60,  or 
at  the  distance  of  the  moon,  requires  sixty  fold  the  time  in  falling 
from  a  state  of  rest  over  a  given  space,  that  a  body  at  the  distance 
1  or  at  the  surface  of  the  earth  does.  Hence  the  time  is  as  the 
distance,  by  any  and  every  consideration  of  the  subject.  Hence, 
if  two  bodies  commence  their  fall  at  the  same  time  at  different 
distances  from  the  centre  of  gravity,  each  to  perform,  describe, 
or  pass  over  a  given  or  like  amount  of  space,  each  being  acted 
upon  by  a  uniform  intensity  of  the  force  of  gravity  equal  to  the 
intensity  by  which  it  commenced  its  fall,  the  rate  or  intensity  of 
gravity  of  each  will  be  to  the  distance  from  the  centre  of  gravity 
at  which  the  fall  commences,  as  it  is  to  the  time  in  which  the 
body  shall  be  falling  over  the  given  space ;  and  this  is  univer- 
sally assented  to.  Hence,  it  is  as  conclusively  determined,  in 
respect  to  falling  bodies,  that  the  time  is  as  the  distance,  as  it  is 
in  respect  to  the  windlass  or  wheel  and  axle  ;  and  in  respect  to 
the  windlass  or  wheel  and  axle,  the  whole  time,  as  also  the  dis- 
tance at  which  ihe  force  or  power  is  applied,  (or  length  of  lever,) 
is  inversely  as  the  rate  or  intensity  of  the  force  applied  at  the 
given  distance. 

If,  then,  the  rate  or  intensity  of  the  force  applied  for  accom- 
plishing the  object,  namely,  the  urging  of  the  body  over  the  given 
space,  be  inversely,  or  in  the  reciprocal  ratio  of  the  time,  it  will 
also  be  in  the  reciprocal  ratio  of  the  distance.  For  as  the  dis- 
tance is  increased,  the  intensity  of  the  force  is  decreased ;  (and 
for  this  reason  alone  is  the  intensity  of  the  force  decreased.)  The 
intensity  of  the  force  is  decreased  in  some  given  inverse  ratio  to 
that  of  time  ;  or  is  decreased  in  some  given  ratio  to  that  of  the 
increase  of  time.  This  ratio  is  found  to  be,  by  every  considera- 
tion which  the  subject  has  yet  received  or  can  receive,  to  be  sim- 
ply the  reciprocal  ratio  of  the  time.  And  hence,  to  repeat,  the 
time  being  as  the  distance,  the  intensity  of  the  force  of  gravity 
decreases  in  the  same  ratio  that  the  distance  increases,  and  not 
in  the  same  ratio  as  the  square  of  the  distance  increases,  as 
taught  by  the  Newtonian  hypothesis.  Let  A  be  conceived  to 
commence  its  fall  at  the  surface  of  the  earth,  or  at  the  distance 
1,  and  we  will  suppose  A  to  fall  over  just  16  feet  of  space  in  one 
second  of  time  (called  one  moment)  from  the  commencement  of 
its  fall ;  and  let  D  commence  its  fall  at  four  semi-diameters  of 
the  earth  from  its  centre,  or  at  the  distance  4 ;  and  the  operation 
to  be  performed  by  D  is  to  fall  from  its  commencement,  or  from 
a  state  of  rest,  over  16  feet  of  space ;  and  with  a  view  to  certain 
expositions,  A  and  D  shall  commence  their  fall  at  the  same  in- 


160  ON    THE    LAW    OF    GRAVITY. 

slant  of  time ;  the  16  feet  over  which  each  body  is  to  fall  being 
called  the  given  space,  which  given  space  is  denoted  by  unity, 
or  is  that  element  in  common  between  the  two  bodies  which  is 
denoted  by  unity.  This  .is  the  task  to  be  performed,  and  we 
wish  to  know  the  proportional  amount  of  labor  required  in  each 
case  to  perform  the  task.  Now  the  16  feet  of  space  to  be  passed 
over  by  either  A  or  D,  is  unity  or  1 ;  the  distance  of  A  is  1 ;  the 
time  of  A  is  1;  the  intensity  or  rate  of  force  applied  to  A  is 
1 ;  the  instant  velocity  of  A,  namely,  the  velocity  of  A  at  any 
given  instant  of  its  fall  while  passing  over  the  given  space,  is  1 ; 
and  the  whole  amount  of  force  applied  to  A  while  passing  over 
the  given  space,  namely,  the  whole  amount  of  labor  bestowed  on 
A  during  its  passage,  is  1.  Hence  each  and  every  element  taken 
into  consideration  when  applied  to  A,  is  assumed  at  unity. 
Hence,  in  respect  to  A,  the  given  space,  being  unity,  is  the  square 
of  the  distance,  of  the  time,  of  the  rate  or  intensity  of  the  force,  of 
the  instant  velocity,  and  of  the  whole  amount  of  force  expended 
on  A  during  the  fall  over  the  givjen  space. 

The  given  space,  to  be  passed  over  by  D,  being  also  unity  or  1, 
is  the  square  of  l-4th  of  the  distance,  or  of  the  time  ;  or  is  l-16th 
of  the  square  of  the  distance  or  of  the  time.  If,  then,  the  intensity 
of  the  force  be  inversely  as  the  square  of  the  distance  or  time, 
then  the  intensity  of  the  force  applied  to  D  will  be  l-16th  of  the 
given  space  ;  and  as  the  time  of  the  application  will  be  4,  hence 
the  whole  amount  of  force  applied  to  D  during  the  fall  over  the 
given  space  will  be  l-4th  of  that  applied  to  A,  and  will  be  denoted 
by  .25,  or  l-4th  of  unity.  But  if  the  intensity  of  the  force  of  gravity 
be  inversely  as  the  distance  or  time,  then  the  intensity  of  the 
force  applied  to  D  will  be  l-4th  as  great  as  that  applied  to  A,  and  as 
the  time  of  the  application  of  the  force  to  D  is  4,  hence  the  whole 
amount  of  force  applied  to  D  during  the  fall  over  the  given  space 
will  be  unity  or  1,  or  just  equal  to  that  applied  to  Awhile  passing 
over  the  given  space.  Let  not,  then,  the  middle  of  a  formula,  ap- 
plied to  D  with  a  view  to  ascertain  the  proportion  which  its  respec- 
tive elements  bear  to  unity  orl,  or  to  like  elements  when  applied 
to  A,  perplex  or  thwart  our  calculations  by  considering  the  uni- 
form series  expressed  by  the  constant  figure  1,  as  only  denoting 
time  or  distance,  or  rather  as  simply  denoting  time. 

The  uniform  series  then  expressed  by  the  constant  figure  1,  as 
well  denotes  equal  quantities  of  force  expended  in  equal  times, 
as  it  does  the  equal  times  into  which  the  whole  time  is  divided. 

The  great  object  of  research,  then,  is  directly  this :  to  ascer- 
tain the  ratio  or  proportion  which  the  whole  amount  of  force 
applied  to  D  while  passing  over  the  given  space,  bears  to  the 


ON    THE    LAW    OF    GRAVITY.  161 

whole  time  of  the  fall  over  the  given  space ;  or  what  is  the  same 
thing,  to  ascertain  the  ratio  of  the  amount  of  force  expended  in 
any  given  portion  of  the  whole  time,  to  that  of  the  given  portion 
of  the  time  in  which  such  given  amount  of  force  shall  have  been 
expended.  But  the  given  space  is  unity,  as  well  when  applied 
to  D  as  when  applied  to  A,  and  it  is  an  easy  matter  to  divide 
the  given  space  into  equal  parts,  so  that  one  of  them  shall  be  per- 
formed by  D  in  the  first  moment,  (or  l-4th  part  of  the  whole  time,) 
three  spaces  in  the  second  moment,  five  in  the  third,  and  seven 
in  the  fourth.  And  such  is  the  common  presentation  and  extent 
of  the  explanation  of  a  numerical  formula  designed  to  exhibit  the 
law  of  a  body  falling  from  a  state  of  rest  towards  the  earth  ;  and 
this  is  in  fact  all  the  exhibition  required  when  but  a  single  falling 
body  is  to  be  considered,  or  when  there  is  no  ratio  to  be  deter- 
mined between  its  elements  and  those  of  another  falling  body 
urged  by  a  greater  or  less  intensity  of  gravity ;  or  rather,  when 
the  intensity  of  gravity  is  not  brought  into  the  calculation  or  in- 
vestigation. But  when  the  intensity  or  rate  of  gravity  becomes 
a  matter  of  consideration  and  inquiry  ;  when  the  rate  or  intensity 
of  gravity  at  different  distances  from  the  centre  of  gravity,  is  to 
be  investigated  by  means  of  a  consideration  of  the  law  of  falling 
bodies,  and  in  which  the  time  (being  as  the  distance)  must  be 
proportioned  to  the  force,  and  which,  consequently,  requires  a 
comparison  of  two  bodies,  commencing  their  fall  at  different  dis- 
tances from  the  centre  of  gravity  ;  —  a  consideration  of  the  formula 
of  D,  which  only  adapts  the  equal  parts  of  unity,  of  which  the 
given  space  is  compared,  to  the  number  of  equal  moments  in 
which  I)  shall  be  falling  over  the  given  space,  will  not  be  likely 
to  lead  to  any  satisfactory  result  in  respect  to  the  ratio  which  the 
intensity  of  gravity  bears  to  the  time  of  the  fall  over  the  given 
space,  and  consequently,  the  ratio  of  the  intensity  of  gravity,  as 
proportioned  to  the  distance. 

The  uniform  series  of  the  formula  for  D,  expressed  by  the  con- 
stant figure  1,  always  denotes  the  distance  at  which  D  commen- 
ces its  fall,  (that  of  A  being  unity; )  it  also  denotes  the  units  of 
time  (called  the  times)  in  which  D  is  passing  over  the  given 
space,  which  is  unity ;  the  time  in  which  A  is  falling  over  the  given 
space,  being  unity  or  1.  So  the  uniform  series  expressed  by  the 
constant  figure  1,  most  emphatically  denotes  the  number  of  equal 
quantities  into  which  the  whole  amount  of  force  expended  on  D 
during  its  fall  over  the  given  space,  is  divided  ;  in  which  case, 
one  of  the  equal  quantities  of  the  whole  force  expended  during 
the  fall,  will  be  expended  in  one  of  the  equal  times ;  for  the  in- 
tensity of  the  force  is  properly  considered  as  uniform  as  the  time. 

Hence,  what  may  properly  be  termed  the  accelerating  series  of 
21 


162  ON    THE    LAW    OF    GRAVITY. 

the  formula,  namely,  the  natural  numbers  in  iheir  order,  or  the 
odd  numbers  in  their  order,  applies  equally  well  to  the  uniform 
series  when  it  denotes  force  as  when  it  denotes  time  ;  arid  the 
sum  of  the  odd  numbers  of  the  accelerating  series  will  be  as  well 
the  square  of  the  forces  into  which  the  whole  force  is  divided,  as 
of  the  times  into  which  the  whole  time  is  divided,  —  as  also  of 
the  distances  into  which  the  whole  distance  is  divided,  (the  tim°s 
and  distances  being  but  so  many  equal  units  of  the  time  and  dis- 
tance of  A.)  Thus  each  of  the  times  or  of  the  distances  denoted 
by  the  uniform  series,  is  unity,  or  is  numerically  equal  to  the  en- 
tire given  space  to  be  passed  over.  Hence,  in  respect  to  time 
and  distance  only,  an  equal  division  of  the  given  space  into  the 
sum  of  the  odd  numbers  in  the  accelerating  series,  would  seem 
to  be  of  but  little  consequence,  so  far  as  the  main  question 
is  concerned  ;  such  being  the  simple  consequence  of  ratio  and 
proportion,  (which  always  accompany  unity  of  purpose,)  the  ben- 
eficial results  of  which  r.re  manifested  by  an  application  of 
such  equal  division  of  the  given  space,  to  an  equal  division  of 
the  whole  amount  of  force  expended  on  the  body  while  pass- 
ing over  the  given  space  ;  in  which  case  the  amount  of  force 
denoted  by  the  first  figure  1,  in  the  uniform  series,  is  expended 
on  the  first  space  in  the  accelerating  series,  the  next  amount  of 
force  upon  the  next  three  spaces,  and  so  on ;  in  which  case 
the  whole  amount  of  force,  whatever  be  its  ratio  to  that  of 
the  time  or  distance,  or  to  unity,  is  expended  in  the  given  space. 
In  such  case  the  amount  of  space  at  any  given  instant  of  time, 
or  at  any  given  point  of  the  space,  generated  by  the  force, 
(not  by  the  time  or  the  distance,)  is  the  square  of  the  whole 
amount  of  force  which  has  been  expended  from  the  commence- 
ment of  the  fall ;  and  consequently  it  is  so  at  the  end  of  the  given 
space,  or  when  the  amount  of  space  generated  has  become  unity. 
Let  D,  then,  commence  its  fall  at  the  distance  4,  and  let  the 
given  space  for  A  and  for  D  be  16  feet,  and  let  them  commence 
their  fall  at  the  same  instant  of  time  ;  then,  when  A  has  fallen 
over  16  feet,  D  will  have  fallen  one  foot.  When  D  has  fallen 
over  16  feet,  its  velocity  will  be  equal  to  that  of  A  when  A  has 
fallen  over  1  foot ;  and  when  D  has  fallen  over  16  feet,  its  velo- 
city will  be  one-fourth  as  great  as  that  of  A  when  A  has  fallen 
over  16  feet.  At  any  given  instant  of  time  while  A  is  falling 
over  the  given  space,  its  velocity  will  be  sixteen-fold  that  of  D  at 
the  same  instant.  At  any  given  point  of  the  given  space,  as  at 
the  end  of  the  first  or  second  foot,  &c.,  the  velocity  of  A  will  be 
four-fold  that  of  D,  at  a  like  point  of  the  given  space;  hence 
D  must  fall  over  16  times  a  given  space,  in  order  to  acquire  the 
velocity  of  A  at  the  end  of  such  given  space.  As  the  time  of  A 


ON    THE    LAW    OF    GRAVITY.  163 

is  1,  and  the  time  of  D  is  4,  and  as  the  whole  time  only  stands 
as  the  reciprocal  of  the  intensity  of  the  force,  waiting  its  opera- 
tion, the  same  amount  of  force  will  be  expended  on  D  as  upon 
A  while  passing  over  the  given  space. 

It  is  easy  to  conceive  upon  what  ground  an  error  might  occur, 
in  an  attempt  to  deduce  the  law  of  gravity  from  a  consideration 
of  the  law  of  falling  bodies,  namely,  by  not  properly  considering 
or  regarding  the  simple  principle  or  axiom,  that  in  a  falling  body 
there  is  no  motion,  not  even  the  least,  without  acceleration ; 
namely,  that  the  whole  motion  is  an  accelerated  motion.  Con- 
sequently in  the  first  moment  of  the  fall,  however  short  we  may 
conceive  such  moment  to  be,  if  it  be  but  time,  the  motion  is  ac- 
celerated therein.  If,  however,  in  the  consideration  of  a  formula, 
we  conceive  the  first  moment  (and  consequently  the  first  space, 
however  short  or  small)  to  be  performed  without  acceleration  of 
motion  ;  and  of  course  upon  a  totally  different  principle  from  that 
which  occurs  in  the  remainder  of  the  time  and  space, — in  such 
case  it  would  at  least  become  problematical  whether  the  intensity 
of  the.  force  of  gravity  should  be  assumed  as  being  inversely  as 
the  square  of  the  distance,  or  inversely  as  the  distance.  For 
nothing  would  occur  in  the  first  moment  or  first  space,  to  make  the 
amount  of  space  then  passed  over,  the  square  of  the  amount  of 
force  then  expended  ;  for  this  principle  arises  wholly  from  the 
•acceleration  of  the  motion.  But  if  there  were  no  acceleration  of 
the  motion  in  the  first  moment  and  space ;  if  in  the  first  space  the 
motion  is  to  be  uniform,  and  only  to  be  accelerated  in  future  ;  then 
it  might  well  be  conceived  that  the  force  is  as  the  motion  in  the  first 
-space  or  first  moment ;  and  consequently  that  in  the  first  space  and 
moment,  the  intensity  of  the  force  of  gravity  is  inversely  as  the 
square  of  the  distance,  or  of  the  whole  time  ;  for  surely,  in  every 
formula,  the  first  space  passed  over  is  but  one  of  the  whole  num- 
ber of  equal  spaces  into  which  the  given  space  is  divided,  which 
makes  the  square  of  the  distance  from  the  centre  of  gravity.  But 
if  such  error  be  once  adopted,  (and  a  more  manifest  error  can 
scarcely  be  conceived,)  the  consequences  and  results  must  ne- 
cessarily partake  of  the  same  error.  And  as  the  force  is  to  be 
uniform  throughout  the  fall  over  the  given  space,  a  like  amount 
of  force  as  that  expended  in  the  first  moment  will  be  expended 
in  the  next  moment  of  equal  length,  and  while  the  body  is  pass- 
ing over  three  times  the  space,  in  consequence  of  an  accelerated 
motion  acquired  in  the  first  moment,  and  while  by  supposition 
the  motion  was  uniform.  The  fact,  however,  is  too  manifest  for 
comment,  that  if  we  assign  some  given  moment  of  time,  as 
one  second  for  instance,  to  the  first  space  in  any  given  formu- 
la, such  moment  being  assumed  at  unity,  the  first  space,  how- 


164  ON    THE    LAW    OF    GRAVITY. 

ever  small,  may  also  be  assumed  at  unity  ;  in  which  case  the  dis- 
tance, the  rate  of  force,  the  rate  of  instant  velocity,  and  the  whole 
amount  of  force  expended  while  passing  over  such  space,  will 
necessarily  be  assumed  at  unity.  Hence  the  first  term  in  what 
is  called  the  uniform  series,  and  the  first  term  in  what  is  called 
the  accelerated  series,  always  form  a  perfect  formula  for  the 
standard  body,  or  the  one  commencing  its  fall  at  the  distance  1, 
by  which  to  compare  the  elements  of  another;  they  being  but 
abstract  numeral  quantities  assumed  at  unity. 

Thus  the  distance  of  the  moon  may  as  well  be  assumed  at 
unity  as  the  surface  of  the  earth,  and  become  the  standard  by 
which  to  compare  another  body  commencing  its  fall  at  sixty 
times  the  distance  of  the  moon  from  the  centre  of  the  earth. 
Nevertheless,  the  motion  of  a  body  commencing  its  fall  at  the 
distance  of  the  moon,  will  be  accelerated  in  its  fall  in  the  first 
second  of  time,  or  in  any  shorter  time ;  and  so  will  that  of 
the  body  commencing  its  fall  at  sixty  times  the  distance  of  the 
moon,  however  small  the  space  may  be  which  is  passed  over  in 
the  first  second  of  time. 

If  then  the  motion  of  the  body  be  accelerated  while  passing 
over  the  first  space,  (and  no  one  will  assume  that  it  is  not,)  then 
the  force  of  gravity  is  inversely  as  the  distance  from  the  centre 
of  gravity. 

Perhaps  a  still  more  simple  exposition  than  has  been  given  of 
the  formulas  denoting  the  law  of  falling  bodies  towards  the  earth, 
may  more  readily  suggest  that  comparison  of  the  elements  to  be 
considered,  from  which  to  deduce  the  force  of  gravity  as  propor- 
tioned to  distance  from  the  centre  of  gravity,  than  any  which  I 
have  heretofore  presented. 

When  two  bodies  are  conceived  to  commence  their  fall  at 
different  distances  from  the  centre  of  the  earth,  or  centre  of  gravi- 
ity,  with  a  view  of  comparing  the  various  elements  of  the  two 
bodies,  they  are  conceived  to  commence  their  fall  at  the  same 
time,  and  one  of  them  (called  the  standard)  is  conceived  to  com- 
mence its  fall  at  the  distance  of  unity,  or  1,  from  the  centre  of 
gravity,  whether  it  commence  its  fall  at  the  surface  of  the  earth 
or  elsewhere,  and  the  other  to  commence  its  fall  at  some  given 
number  of  units  from  the  centre  of  gravity,  as  at  the  distance  4, 
8,  60,  or  any  other  given  distance  ;  and  the  space  to  be  perform- 
ed or  passed  over  by  each,  as  the  entire  operation  of  each,  is  to 
be  equal  one  with  the  other,  and  is  assumed  at  unity,  or  1,  in 
respect  to  either  body.  Such  space  is  called  the  given  space ; 
and  may,  if  any  prefer  to  consider  it  in  our  arbitrary  measures  in 
preference  to  abstract  numeral  quantities,  be  assumed  at  16  feet; 
and  that  the  body  at  the  distance  1,  will  pass  from  a  state  of  rest 


ON    THE    LAW    OF    GRAVITY.  165 

over  the  given  space,  in  consequence  of  the  force  of  gravity,  in 
one  second  of  time,  permitting  a  second  of  time  to  be  called  a 
moment,  when  used  in  the  investigation,  which  moment  is  to  be 
understood  as  being  one  second  of  time. 

As  each  and  every  element  of  the  body  which  commences 
falling  from  the  distance  1,  is  properly  or  necessarily  assumed  at 
unity,  hence  the  formula  for  such  body  can  only  consist  of  two 
figures,  each  denoting  unity,  thus : 

1 

the  lower  figure  denoting  the  given  space  to  be  passed  over,  and 
the  upper  figure  denoting  the  second,  or  moment  of  time,  in 
which  the  space  is  performed  ;  also  the  rate  or  intensity  of  the 
uniform  force  with  which  the  body  is  urged,  as  also  the  whole 
amount  of  force  applied  to  the  body  while  passing  over  the  giv- 
en space,  or  the  given  moment  of  time.  If  a  body  commence 
its  fall  at  the  distance  of  4  from  the  centre  of  gravity,  its  formula 
will  necessarily  be  thus  : 

1111 

1234567 

In  this  formula  the  given  space  is  divided  into  sixteen  equal 
parts,  denoted  by  the  sum  of  the  odd  numbers  contained  in  the 
lower  or  accelerated  series ;  while  the  time  of  the  passage  of  the 
body  over  the  given  space  is  increased  to  four  units,  denoted  by 
the  four  figures  in  the  upper  series ;  that  is,  the  time  being  as  the 
distance,  is  in  this  formula  increased  to  four  seconds,  or  mo- 
ments. So  also  the  upper  series  may  denote  an  equal  division 
of  the  whole  amount  offeree  expended  during  the  fall  over  the 
given  space,  into  four  equal  parts  ;  the  first  part  to  be  expended 
in  the  first  moment,- the  second  part  in  the  second  moment,  and 
so  on.  Hence  one  fourth  of  the  whole  amount  of  force  will  be 
expended  in  one  fourth  of  the  whole  time  ;  and  one  foot  of  the 
given  space,  or  one  sixteenth  of  the  given  space,  will  be  passed 
over  in  the  first  moment,  or  in  one  fourth  of  the  whole  time ; 
three  feet  in  the  second  moment,  five  feet  in  the  third,  and  seven 
in  the  fourth,  in  which  case  the  number  of  equal  spaces  into 
which  the  given  space  is  thus  divided,  is  the  square  of  the  num- 
ber of  seconds  or  moments,  or  units  of  time  in  which  the  body 
is  falling  over  the  given  space,  or  over  unity,  as  also  the  square 
of  the  number  of  equal  parts  into  which  the  whole  amount  of 
force  expended  during  the  fall  is  divided.  It  is  then  the  whole 
amount  of  the  force  expended  while  the  body  is  falling  over  the 
given  space,  denoted  by  unity,  that  we  are  seeking,  and  conse- 
quently we  wish  to  obtain  its  rate  or  intensity  at  the  distance  4, 
compared  with  that  at  the  distance  1,  or  compared  with  unity. 


166  ON    THE    LAW    OF    GRAVITY. 

Thus  if  at  the  distance  4  the  rate  be  but  l-16th  of  unity,  then  it 
is  inversely  as  the  square  of  the  distance.  If  at  the  distance  4, 
its  rate  be  l-4th  of  unity,  then  it  is  inversely  as  the  distance. 
Thus  at  the  distance  4,  the  rate  of  force  is  in  some  given  in- 
verse ratio  to  the  whole  distance,  or  whole  time  ;  and  whether  it 
be  in  the  inverse  ratio  to  the  first  or  second  power  of  the  whole 
time,  or  whole  distance,  (for  the  time  is  as  the  distance,)  is  the 
question. 

It  will  be  readily  seen  that  the  formula  for  a  body  that  com- 
mences its  fall  at  the  distance  4,  is  a  perfect  epitome  of  a  formu- 
la for  a  body  commencing  its  fall  at  a  greater  distance  than  1. 
Nevertheless  if  the  formula  be  ever  so  far  extended,  as  for  in- 
stance, if  the  body  be  conceived  to  commence  its  fall  at  the  dis- 
tance 60.  in  which  case  the  uniform  series  expressed  by  the  con- 
stant figure  l,will  be  extended  to  sixty  places,  (denoting  the  num- 
ber of  unit  distance  s,  and  also  the  number  of  seconds  or  unit 
times,)  still  the  space  to  be  passed  over  is  unity,  notwithstanding 
it  is  divided  into  as  many  equal  parts  as  unity  is  contained  in 
the  sum  of  the  odd  numbers  found  in  the  second  or  accelerating 
series,  namely,  3600,  if  the  distance  or  number  of  seconds  or 
moments  be  60.  The  accelerating  series  then  divides  unity 
into  equal  parts,  but  does  not  increase  or  decrease  it,  while  the 
uniform  series  denotes  the  increase  of  the  units  of  time  from  that 
of  one  only,  which  is  required  when  the  distance  is  1,  to  that  of 
the  number  of  figures  in  the  uniform  series.  Each  figure,  then, 
in  the  uniform  series,  so  far  as  it  refers  to  time  or  distance,  is 
unity. 

The  left  hand  figure,  or  figure  1,  in  the  accelerating  series,  al- 
ways denotes  an  amount  of  space  equal  to  the  reciprocal  of  the 
square  of  the  distance,  or  of  the  time  in  which  the  body  is  pass- 
ing over  the  given  space  or  the  whole  space  to  be  passed 
over,  namely,  unity ;  nevertheless,  the  time  of  passing  over  the 
amount  of  space  denoted  by  figure  1  in  the  accelerating  series, 
is  unity,  however  small  such  space  may  be;  and  it  is  manifest 
that  by  extending  the  distance  such  space  may  become  exceed- 
ingly small.  Nevertheless,  if  in  the  first  moment  of  the  fall  there 
be  motion,  (and  the  fall  implies  motion,)  then  space  is  passed 
over  in  the  first  moment  of  time,  or  in  one  second  of  time,  how- 
ever small  such  space  may  be. 

Now  it  may  be  set  down  as  a  lemma,  or  even  an  axiom,  that 
in  a  falling  body  there  is  no  motion  without  a  corresponding  ac- 
celeration of  the  motion;  and  that  the  same  force  which  causes 
the  motion,  at  the  same  time  causes  the  corresponding  accelera- 
tion of  the  motion.  Hence  if  a  body  commence  its  fall  from 
ever  so  many  units  of  distance,  and  consequently  the  unit  of 


ON    THE    LAW    OF    GRAVITY.  167 

space  to  be  passed  over  be  divided  into  ever  so  many  equal 
parts,  (the  number  of  which  is  always  the  square  of  ihe  number 
of  units  of  distance  or  of  time,)  the  motion  of  the  body  while 
passing  over  the  first  equal  part  of  the  given  or  unit  space,  name- 
ly, that  denoted  by  the  left  hand  figure  in  the  accelerating  series, 
will  be  constantly  accelerated. 

Hence  the  uniform  series,  expressed  by  ihe  constant  figure  1, 
may  be  used  to  denote  the  rate  or  intensity  of  the  force,  not  as 
units  of  force,  as  it  does  units  of  time  or  of  distance,  but  in  some 
given  reciprocal  ratio  to  that  of  the  whole  lime  or  the  whole  dis- 
tance, as  that  of  the  simple  reciprocal  ratio,  or  as  the  square  of 
the  reciprocal  ratio  of  the  whole  time  or  whole  distance. 

If  the  rate  of  force  be  in  the  simple  reciprocal  ratio  of  the 
whole  time  or  wThole  distance,  namely,  inversely  as  ihe  whole 
time  or  whole  distance,  then  the  whole  amount  of  force  expend- 
ed while  the  body  is  falling  over  the  given  space,  namely,  unity 
of  space,  will  be  unity  of  force  ;  that  is,  the  whole  amount  of  force 
expended  will  be  unity,  and  the  amount  of  space  passed  over 
will  be  the  square  of  the  whole  amount  of  the  force  then  expend- 
ed. In  such  case  the  motion  of  the  body  will  have  been  accele- 
rated, throughout  the  whole  given  space,  as  well  in  that  part  de- 
noted by  the  first  figure  in  the  accelerating  series,  (however  small 
or  great,  whether  it  be  the  l-16th  part  or  the  l-3600th  part,)  as  in 
any  other  part  of  the  given  space.  That  is,  at  any  point  of  the 
given  space,  or  at  ariy  instant  of  the  fall,  the  amount  of  space  then 
passed  over,  however  small  a  portion  of  unity  it  may  be,  will  be 
the  square  of  the  amount  of  force  which  has  been  expended.  But 
if  the  rate  or  intensity  of  the  force  be  the  reciprocal  ratio  of  the 
square  of  the  whole  time  or  of  the  whole  distance,  namely,  in- 
versely as  the  square  of  the  whole  time  or  of  the  whole  distance, 
then  the  whole  amount,  of  force  expended  on  the  body  while 
passing  over  the  given  space,  namely,  over  unity  of  space,  will 
be  the  simple  reciprocal  ratio  of  the  whole  lime,  or  whole  dis- 
tance, (thus  giving  to  the  whole  amount  of  force  expended  dur- 
ing the  fall,  the  same  numerical  quantity,  or  the  same  ratio  com- 
pared with  unity,  that  should  be  given  to  the  intensity,)  in  which 
case,  at  the  distance  60,  the  amount  of  force  required  to  urge  the 
body  over  the  given  space,  would  be  but  l-60th  of  what  would 
be  required  at  the  distance  1. 

Now,  a  more  manifest  proposition  can  scarcely  be  stated  or 
considered,  than  that  such  a  principle  wholly  prohibits  accelera- 
tion to  the  motion  of  the  body  while  passing  over  the  first  space, 
or  that  denoted  by  the  first  figure  in  the  acceleraling  series,  now- 
ever  great  or  small  such  space  may  be,  and  only  permits  accele- 
ration to  take  place  in  future  ;  and  hence  the  first  space  passed 


168  ON    THE    LAW    OF    GRAVITY. 

over  is,  by  such  a  course  of  reasoning,  treated  in  the  same  man- 
ner as  the  given  space  passed  over  by  the  body  falling  from 
the  distance  1,  and  in  the  time  1,  and  in  which  the  intensity 
of  the  force,  as  also  the  whole  amount  of  force,  are  respective- 
ly assumed  at  unity,  and  in  which  the  element  of  acceleration  is 
not  necessarily  an  element  to  be  considered.  If,  then,  the  motion 
over  the  first  space  be  not  accelerated,  but  uniform,  then  the 
intensity  of  the  force  while  the  body  is  passing  over  the  first 
space,  (however  great  or  small,)  is  as  the  motion  of  the  body 
moving  ;  but  if  the  motion  be  accelerated,  then  the  intensity  of 
the  force  applied  is  not  as  the  motion  of  the  moving  body.  The 
consequence  then  is  most  obvious ;  for  if  we  assume  the  motion 
to  be  uniform  in  the  first  space,  however  small,  we  necessarily 
make  the  intensity  of  the  force  inversely  as  the  square  of  the 
time,  or  of  the  distance ;  and  consequently  the  whole  amount  of 
force  expended  while  the  body  is  passing  over  unity,  or  the  giv- 
en space,  will  be  the  inverse  of  the  distance,  or  of  the  time. 

For  the  amount  of  force  expended  in  the  first  space,  (however 
small  such  space  may  be,)  is  the  standard  or  criterion  by  which 
to  compare  the  future  acceleration ;  for  the  intensity  of  the  force 
being  uniform,  a  like  amount  of  force  to  that  expended  in  the 
first  space  will  be  expended  in  the  three  succeeding  spaces,  or  in 
the  next  five,  and  so  on. 

Hence  the  advocates  of  such  hypothesis  must  say  that  the 
same  amount  of  force  which  produces  a  uniform  motion  over 
the  first  space,  (however  great  or  small  such  space  may  be,)  will 
produce  an  accelerated  motion  over  the  three  succeeding  spaces, 
—  and  so  on.  The  simple  consequence  then  is,  if  the  motion 
in  the  first  space  be  uniform,  the  rate  or  intensity  of  the  force  will 
be  inversely  as  the  square  of  the  distance.  But  if  there  be  no  part 
of  the  given  space  in  which  the  motion  is  uniform,  or  not  ac- 
celerated, or  in  which  the  rate  or  intensity  of  the  force  applied 
is  as  the  motion  of  the  falling  body,  then  the  ratio  or  intensity 
of  the  force  applied  is  inversely  as  the  whole  distance,  or  the 
whole  time  of  the  fall  over  the  given  space. 

Mr.  Vince  says  that  "  Sir  Isaac  Newton  found  that  if  the 
force  with  which  bodies  fall  upon  the  earth's  surface  were  ex- 
tended to  the  moon,  and  to  vary  inversely  as  the  square  of  the 
distance  from  the  centre  of  the  earth,  it  would  in  one  minute 
draw  the  moon  through  a  space  which  is  equal  to  the  versed 
sine  of  the  arc  which  the  moon  describes  in  one  minute.  He 
concluded,  therefore,  that  the  moon  was  retained  in  its  orbit  by 
the  same  force  as  that  by  which  bodies  are  attracted  upon  the 
earth."  That  the  power  of  attraction,  which  causes  bodies  to 
fall  to  the  earth,  extends  to  the  moon,  and  retains  it  in  its  orbit, 


ON    THE    LAW    OF    GRAVITY.  169 

is  perhaps  too  manifest  for  any  one  to  doubt ;  and  this  was  the 
conclusion  of  Kepler  long  before  the  lime  of  Newton.  But  how 
Sir  Isaac  Newton  should  have  come  to  such  a  conclusion  from 
his  observaiions  and  calculations  may  appear  somewhat  mys- 
terious. 

Now  that  the  versed  sine  of  the  arc  which  the  moon  would 
d  scribe  in  its  orbit  in  60  seconds  of  time,  is  equal  to  the 
amount  of  deflection  from  a  tangent  in  the  same  time,  all  will 
agree.  This  part  of  the  subject  I  have  before  partially  ex- 
amined; arid  the  reason  for  preferring  the  method  of  consider- 
ing the  amount  of  deflection  from  the  tangent,  in  preference  to 
the  versed  sine  of  the  arc,  is  because  it  gives  a  much  more  com- 
prehensive range  for  investigation, 

Now  I  will  not  undertake  to  say  but  that  the  versed  sine  of 
the  arc  passed  over  by  the  moon  in  60  seconds,  is  equal  to  the 
fall  of  a  body  from  a  state  of  rest  near  the  earth  in  one  second ; 
to  be  sure  the  space  passed  over  in  60  seconds  is,  in  respect  to 
the  circumference  of  the  moon's  orbit,  a  very  little  distance ; 
and  so  far  as  could  be  well  ascertained  from  observation  alone, 
would  not  present  an  error  that  could  readily  be  detected  by  ob- 
servation. Nevertheless,  such  little  distance  would  be  sufficient  to 
contain  a  mathematical  error,  vast  in  its  consequences,  if  it  ex- 
isted. But  it  must  be  presumed  that  Sir  Isaac  Newton  drew  his 
conclusion,  that  the  attraction  of  the  earth  extended  to  the  moon, 
from  some  calculation  made  upon  said  observations ;  for  other- 
Wise,  he  might  as  well  have  supposed  that  Jupiter  or  Saturn 
was  retained  in  its  orbit  by  the  attraction  of  the  earth.  But 
Mr.  Vince  says  that  Sir  Isaac  Newton  found,  &c.,  but  the  el- 
ements upon  which  he >  found  are  not  given  by  Mr.  Vince,  and 
hence  it  is  fair  to  conclude  that  they  are  the  same  that  are  given 
in  other  places,  by  which  it  is  said  the  law  of  gravity  was  dis- 
covered to  be  inversely  as  the  square  of  the  distance,  viz :  that 
the  moon  being  60  times  as  far  from  the  centre  of  the  earth  as 
the  surface  is,  would  be  deflected  as  far  from  a  tangent  to  its 
orbit,  in  60  seconds  of  time,  as  a  body  near  the  surface  of  the 
earth  would  fall  from  a  state  of  rest  in  one  second  of  time ; 
and  that  this  amount  of  deflection  from  a  tangent  would  be 
equal  to  the  versed  sine  of  the  arc  of  the  moon's  orbit  which  it 
would  pass  over  in  60  seconds,  and  which  induced  the  conclusion, 
not  only  that  the  attraction  of  the  earth  extends  to  the  moon,  but 
that  it  varies  inversely  as  the  square  of  the  distance.  Hence,  the 
statement  of  Mr.  Vince  may  be  resolved  into  this:  that  Sir  Isaac 
Newton,  according  to  his  calculation,  found  that  if  gravity  de- 
creases inversely  as  the  square  of  the  distance,  that  a  body  fall- 
ing from  a  state  of  rest  directly  towards  the  earth,  at  the  dis- 
22 


170  ON    THE    LAW    OF    GRAVITY. 

tance  of  the  moon,  would  fall  as  far  in  60  seconds,  as  a  body 
at  the  surface  of  the  earth  would  in  one  second,  and  that  the 
amount,  of  such  fall  in  60  seconds  would  be  equal  to  the  versed 
sine  of  the  arc  which  the  moon  would  pass  over  or  describe  in 
the  same  time;  or  equal  to  the  deflection  of  the  moon  from  a 
tangent  in  the  same  time  ;  and  we  must  add  also,  that  the  moon 
is  60  times  as  far  from  the  centre  of  the  earth  as  the  surface  of 
the  earth  is. 

Now  that  there  is  error  in  all  these  positions  except  that  of 
the  distance  of  the  moon  from  the  centre  of  the  earth,  may  per- 
haps be  safely  alleged  upon  the  truth  of  mathematics,  geometry 
and  the  law  of  falling  bodies. 

I  am  aware  that  Mr.  Vince's  statement  of  what  he  and  others 
suppose  Sir  Isaac  Newton  to  have  found,  is  rather  vague  and 
intangible;  and  that  if  a  proposition  be  drawn  from  it  for  the 
purpose  of  demonstration,  either  by  way  of  reductio  ad  absur- 
dum,  or  otherwise,  certain  things  not  contained  in  the  statement 
of  Mr.  Vince,  must  be  contained  in  the  proposition  ;  as  that  the 
moon  is  sixty  times  as  far  from  the  centre  of  the  earth  as  the  sur- 
face of  the  earth  is.  Then  the  proposition  may  be  put  thus,  viz: 
If  gravity  decreases  from  the  centre  ol  the  earth  to  the  distance  of 
the  moon,  inversely  as  the  square  of  the  distance,  and  a  body  at 
the  surface  of  the  earth  will  fall  from  a  state  of  rest  16  feet  in  one 
second  of  time,  how  far  will  a  body,  at  the  distance  of  the  moon, 
fall  from  a  state  of  rest,  in  60  seconds ;  or  how  far  will  the  moon 
be  deflected  from  a  tangent  to  its  orbit  in  bO  seconds  ?  And  it  ap- 
pears to  me  that  this  proposition  is  so  far  based  upon  the  state- 
ment of  Mr.  Vince,  as,  when  solved,  to  determine  what  Sir 
Isaac  Newton  found,  unless  he  found  something  that  does  not 
exist.  Inasmuch  as  the  dimensions  of  what  he  found  are  not 
given  by  Mr.  Vince,  either  in  length  or  breadth,  we  must  en- 
deavor to  deduce  them  as  well  as  we  can  from  the  premises  ; 
and  first,  a<  to  the  distance  a  body  would  fall  in  60  seconds, 
from  a  state  of  rest,  at  the  distance  of  the  moon,  or  at  distance 
60,  upon  the  assumption  that  gravity  varies  inversely  as  the 
square  of  the  distance  from  the  centre  of  the  earth  to  the  dis- 
tance of  the  moon  ;  and  that  a  body  falling  from  a  state  of  rest 
at  the  surface  of  the  earth  will  fall  16  feet  in  one  second  of  time  ; 
and  also  that  the  amount  of  fall  at  the  distance  of  the  moon  is 
to  be  determined  by  the  law  of  falling  bodies.  Upon  the  New- 
tonian law  of  gravity,  then,  a  body  falling  from  a  state  of  rest, 
at  the  distance  of  the  moon,  would  fall  in  60  seconds  of  time 
the  1-. 3600th  part  of  16  feet,  or  a  little  more  than  l-20ih  of  an 
inch;  in  which  case  it  would  fall  in  the  first  second  of  time  the 
l-12,600,000th  part  of  16  feet.  It  is  a  very  little  distance,  to  be 


ON    THE    LAW    OF    GRAVITY.  171 

sure,  compared  with  the  time,  but  it  is  all  that  the  law  of  falling 
bodies  will  allow  upon  the  Newtonian   hypothesis  of  the  law  of 
gravity.    This  quantity  then,  viz:  the  1  -it3<  0  h  part  of  16  feet,  ac- 
cording to  said  proposition,  would  be  Sir  Isaac  Newton's  versed 
sine  of  the  arc  which  the  moon  would   describe  in  60  seconds; 
and  consequently,  according  to  his    calculation,  would    be  the 
amount  of  deflection  of  the  moon  from  a  tangent  in  the  same 
time.      Nevertheless,    a   body  at   the  distance   60  will   fall  the 
l-3600th  part  of  16  feet  in  the  first  second.     Let  us  now  sup- 
pose that  a  body  at  the  surface  of  the  earth,  viz:  at  the  distance 
1,  will  fall  16  feel  from  a  state  of  rest  in   one  second  of  time  ; 
and  that  a  body  at  the  distance  of  the  moon,  viz:  at  the  distance 
60,  will  fall  16  feet  from  a  state  of  rest  in   60  seconds  of  time, 
which  are  the  true  proportions.     This  proposition  will   at  once 
shift  the  supposed  law  of  gravity  from  the  inverse  of  the  square 
of  the  distance,  to  the  inverse  of   the  distance,  or  to  the  law  as 
it  is,  if  it  be  in    anywise  corroborated    by  the    law  of  falling 
bodies.     In  such  case,  the  amount  of  fall  at  the  distance  60  in 
the  first  second  of  time,  will  be  the  1 -3600th  part  of  16  feet;  in 
the  next  second,  the  l-1200th  part;  in   the  third  second  of  time, 
the  l-720th  part  of  16  feet,  and   so  on ;  thus  increasing  in  each 
successive  second  of  time   in  the  ratio  of  the  next  succeeding 
odd   number ;  and   in  which  case,  the  whole  16  feet  would  be 
accomplished  in  the  60  seconds,  in  accordance  with   the  law  of 
falling  bodies.     But  no  one  can  rationally  suppose  that  gravity 
(which  is  as  uniform    at   any  given  distance   as  time   is)  will, 
at  the  distance  60,  exert  three  times  the  force  in  the  second  sec- 
ond of  time  that  it  does  in   the  first ;  five  times  as  much  in  the 
third,  and  so  on,  any  more  than  it  will  at  the  distance  1,  or  at 
the  surface  of  the  earth;  or  than  time  will;  which  it  must  do, 
if  gravity  be  inversely  as  the  square  of  the  distance,  in  order  that 
a  body  at  the  distance  60  may  fall  as  far  in  time  60,  as  a  body 
at  the  distance  1,  will    fall  in  time  1.     But  this  is  palpably  plain, 
and  it  being  within  the  reach  of  every  person  to  examine  and  as- 
certain the  truth  for  himself,  mankind  should  not  have  been  lelt 
in  an  error  in  respect  to  it  for  a  moment. 

I  will  acknowledge  that  in  respect  to  the  extent  of  what  it  is 
supposed  Sir  Isaac  Newton  found,  it  is  left  somewhat  vague 
and  intangible  by  Mr.  Vince.  Nevertheless,  mankind  have  got 
the  idea,  or  notion,  that  he  found  that  the  moon  will  be  deflected 
as  far  from  a  tangent  in  60  seconds  of  time,  as  a  body  would 
fall  from  a  state  of  rest  in  the  same  time,  at  the  distance  of  the 
moon.  For  it  is  said  that  the  moon  would  be  drawn  towards 
the  earth  by  gravity,  in  60  seconds,  through  the  versed  sine  of 
the  arc  that  the  moon  would  describe  in  that  time  in  her  orbit. 


172  ON    THE    LAW    OF    GRAVITY. 

This  is  a  serious  allegation  ;  and  when  taken  implicitly,  and 
without  examination  to  see  if  these  things  are  so,  or  can  be  so,  is 
well  calculated  to  lead  into  a  wilderness  of  error,  in  lieu  of  lead- 
ing to  the  truth  in  matters  depending  upon  such  premises.  And 
I  am  confident  the  error  contained  in  I  he  above  statement  of 
what  Sir  Isaac  Newton  found,  has  been  loo  long  continued  for 
the  benefit  of  philosophy. 

The  rate  of  convergency  of  a  planet  is  inversely  as  the  period. 
What  may  be  called  deflection  in  time,  (so  far  as  rate  may  be 
said  to  apply  to  deflection.)  is  inversely  as  the  square  of  the  dis- 
tance, or  equal  to  Sir  Isaac  Newton's  rate  of  force.  And  the 
rate  of  deflection  in  space  is  inversely  as  the  distance,  or  the 
square  root  of  the  rate  of  deflection  in  time.  Hence  the  rate  of 
convergency  is  a  mean  proportional  between  the  rate  of  deflec- 
tion in  space  and  the  rate  of  deflection  in  time.  But  as  the 
factitious  element  of  deflection  in  time  and  space  requires  a 
more  labored  explanation,  we  will  not  dwell  upon  it  here.  But 
I  will  here  make  the  allegation  that  no  one  will  deny,  namely, 
that  the  greater  the  mean  distance  of  a  planet  of  the  same  sys- 
tem is,  the  greater  will  be  the  motion  in  proportion  to  the  rate  of 
gravity  or  force;  and  the  less  will  be  the  rate  of  convergency  in 
proportion  to  the  rate  of  gravity.  For  it  is  only  upon  this  prin- 
ciple that  the  squares  of  the  periods  of  a  system  of  planets  can 
be  as  the  cubes  of  their  mean  distances  from  the  sun.  Thus, 
upon  the  hypothesis  that  gravity  is  inversely  as  the  distance,  the 
mean  rate  of  gravity  will  always  be  a  mean  proportional  be- 
tween the  mean  rate  of  motion,  and  the  mean  rate  of  conver- 
gency ;  and  in  the  same  ratio  that  motion  increases  in  proportion 
to  the  rate  of  force  as  the  distance  increases,  so  the  rate  of  con 
vergency  decreases  in  proportion  to  the  rate  of  gravity.  Thus 
at  the  distance  4  the  rate  of  motion  is  twice  as  s^reat  in  propor- 
tion to  the  rate  of  gravity  as  at  the  distance  1  ;  and  the  rate  of 
convergency  is  half  as  great  in  proportion  to  the  rate  of  gravity 
as  at  the  distance  1.  That  is,  by  the  hypothesis  that  gravity  is 
inversely  as  the  distance,  the  rate  of  motion  increases  upon  the 
rate  of  gravity  as  the  square  root  of  the  distance  increases  from 
unity ;  and  the  rate  of  convergency  decreases  upon  gravity  as 
t  e  rate  of  motion  decreases  from  unity. 

But  upon  the  hypothesis  that  gravity  is  inversely  as  the  square 
of  the  distance,  the  rate  of  motion  will  increase  upon  the  rate  of 
gravity  as  the  period  increases  from  unity.  Thus  if  the  dis- 
tance be  4,  the  rate  of  motion  will  be  eight  times  the  rate  of 
gravity.  And  upon  such  hypothesis  of  gravity,  the  rate  of  con- 
vergency will  increase  upon  the  rate  of  gravity  as  the  distance 
increases,  in  the  same  ratio  that  the  square  root  of  the  distance  in- 


ON    THE    LAW    OF    GRAVITY.  173 

creases  from  unity;  so  that  when  the  rate  of  motion  is  eight  times 
the  rate  of  gravity,  the  rate  of  convergeney  will  be  twice  the  rate 
of  gravity.  Hence  the  mean  rate  of  motion  is  to  the  mean  rate 
of  convergeney  as  the  mean  distance  is  to  unity,  by  any  law  of 
gravity  ;  nor  do  the  mean  rales  of  motion  and  convergeney,  as 
proportioned  to  each  other  and  to  the  distance  and  period,  de- 
pend upon  our  knowledge  of  their  proportions,  or  upon  our  know- 
ing how  gravity  is  proportioned  to  distance.  But  the  reason  for 
introducing  the  foregoing  principles  in  respect  to  the  proportion 
between  motion  and  convergeney,  is  for  the  purpose  of  showing 
that  the  motion  of  a  planet  in  its  orbit  counteracts  conver- 
geney ;  that  is,  if  the  motion  be  increased  in  proportion  to  the 
force  or  gravity,  the  rate  of  convergeney  will  fall  short  of  the 
rate  of  motion  in  the  ratio  of  the  distance  to  unity  by  either  hy- 
pothesis of  force.  And  hence,  as  the  distance  is  increased,  the 
motion  becomes  greater,  in  proportion  to  the  force,  and  also 
greater  in  proportion  to  the  convergeney  by  either  hypothesis  of 
gravity.  So  then  at  the  distance  4  the  convergeney  of  a  body  will 
not  be  as  great  in  proportion  to  the  motion  as  when  the  distance 
is  1.  The  consequence  then  is  apparent,  that  as  the  distance  is 
increased,  not  only  the  convergeney  but  the  deflection  is  less  in 
proportion  to  the  motion.  This  becomes  apparent  if  we  take  a 
planet  \\hose  distance  is  1,  and  another  whose  distance  is  4,  and 
conceive  the  planet  whose  distance  is  1,  to  have  passed  over  l-4th 
of  its  orbit,  and  the  planet  whose  distance  is  4,  to  have  passed  over 
l-8th  of  its  orbit,  or  over  twice  the  space  of  the  other,  and  in  quad- 
ruple the  time  ;  in  which  case,  the  deflection  of  the  planet  whose 
distance  is  1,  is  unity,  while  the  deflection  of  that  whose  distance 
is  4,  is  1.171576,  in  which  case,  if  deflection  were  to  proceed 
upon  the  principle  of  falling  bodies,  when  the  time  became 
doubled,  —  namely,  when  the  planet  whose  distance  is  4  had  pass- 
ed over  l-4th  of  its  orbit,  —  the  amount  of  deflection  would  be 
4.686304,  when  in  fact  it  is  but  4.  Hence  the  deflection  was  not 
three  times  as  great  in  the  last  half  of  the  time  as  it  was  in  the 
first;  and  hence  deflection  does  not  correspond  with  the  law  of 
falling  bodies. 

I  have  made  the  example  broad  for  the  purpose  of  showing 
the  error  in  the  expanse.  Nor  is  the  error  mathematically  avoid- 
ed, by  supposing  a  very  little  time  to  have  been  passed  over;  as 
that  passed  over  in  60  seconds  of  time  by  the  moon.  And 
hence,  Sir  Isaac  Newton  did  not  find,  by  any  law  of  gravity 
whatever,  that  the  versed  sine  of  any  small  arc  of  the  moon's 
orbit  would  be  the  distance  over  which  a  body  would  fall  from  a 
state  of  rest  at  the  distance  of  the  moon,  while  the  moon  was 
describing  or  passing  over  such  arc.  We  hence  find  that  the 
deflection  of  a  planet,  or  fall  from  a  tangent,  is  not  likened  in  its 


174  ON    THE    LAW    OF    GRAVITY. 

progression  to  that  of  falling  bodies;  and  hence,  as  an  element 
for  deducing  the  law  of  gravity,  it  is  illy  qualified,  being  too  for- 
eign and  factitious.  It  may,  however,  be  proper,  inasmuch  as  so 
much  reliance  has  heretofore  been  placed  upon  it,  to  give  it  still 
further  consideration,  in  which  not  only  its  futility  as  such  ele- 
ment may  be  shown,  but  many  examples  given  by  which  it  will 
be  shown  that  most  of  the  deductions  which  are  apparently 
drawn  from  its  consideration  are  mere  fictions,  and  not  reali- 
ties. This  is  the  case  in  a  previous  section,  notwithstanding  it 
presents  much  imporlant  truth  ;  and  such  will  consequently  be 
the  case  in  the  remainder  of  the  present  section  so  far  as  deflec- 
tion is  treated  of  upon  the  principle  that  it  is  one  of  the  legiti- 
mate elements  of  the  planets  and  their  orbits;  and  hence  the 
reader  must  not  consider  it  as  an  element  proper,  but  that  most 
of  the  apparent  results  drawn  from  its  consideration  are  fictions; 
however  much  it  may  serve  as  a  discipline  designed  to  exhibit 
specious  errors,  for  the  purpose  of  avoiding  them. 

I  have  said  that  deflection  may  be  properly  divided  into  deflec- 
tion in  time  and  deflection  in  space ;  and  that  the  rate  of  deflec- 
tion in  time,  is  the  square  of  the  rate  of  deflection  in  space. 
For  example:  Take  the  planets  A  and  C, —  A  at  the  dis- 
tance 1,  and  C  at  the  distance  4,  and  let  them  pass  from  their 
tangents  some  little,  but  equal  time;  in  which  case  C,  will  have 
passed  over  one  half  the  space  that  A  has;  and  according  to  the 
commonly  received  notions  of  deflections,  (which,  however,  we 
have  shown  to  be  erroneous,)  A  will  have  fallen  sixteen  times 
as  far  from  its  tangent  as  C  has.  And  such  is  what  I  have 
thought  proper  to  call  deflection  in  time. 

Suppose  A  should  stop,  and  C  pass  on  over  equal  space  in  its 
orbit  that  A  has,  or  twice  the  time  that  A  was  passing  over  equal 
space,  (which  I  have  thought  proper  to  call  the  deflection  in  space, 
or  the  deflection  of  two  planets  while  passing  over  equal  space.) 
Thus,  if  it  were  the  case,  that  in  equal  time,  the  deflection  of  C 
was  l-16th  that  of  A,  and  while  passing  over  equal  space  was 
l-4th  that  of  A,  the  deflection  of  C  in  time,  would  be  the  square  of 
its  deflection  in  space.  Hence,  I  have  made  the  allegation  upon 
such  hypothesis ;  and  in  such  case,  the  deflection  in  time  would 
be  the  inverse  of  the  square  of  the  distance,  while  the  deflection 
in  space  would  be  the  inverse  of  the  distance. 

If  the  deflection  of  C  at  the  end  of  equal  time,  were  l-16th  of 
that  of  A,  and  if  its  fall  from  a  tangent  were  governed  by  the 
law  of  falling  bodies,  then,  at  the  end  of  equal  space,  its  deflec- 
tion must  necessarily  be  l-4th  that  of  A.  But  we  have  shown  that 
such  are  not  the  facts.  It  is  then  but  simple  hypothesis,  based 
upon  the  erroneous  supposition  that  deflection  is  identical  in  its 
construction  with  that  of  falling  bodies. 


ON    THE    LAW    OF    GRAVITY.  175 

We  thus  perceive  that  we  are  dealing  in  hypothesis,  the  re- 
sults of  which  are  fictions,  and  not  those  realities  which  are 
material  to  the  issue;  or  at  least  those  fictions  are  but  collateral 
to  the  issue.  But  let  us  pursue  a  liitle  the  consequences  arising 
from  the  foregoing  hypothesis;  and  to  this  end  I  will  take  A, 
and  another  planet,  at  distance  60,  denoted  by  D,  —  or  a  body 
revolving  at  the  surface  of  the  earth,  and  another  at  the  distance 
of  the  moon. 

Now  their  relative  motions  will  be  as  1  to  .12909,  or  inversely 
as  the  square  roots  of  their  respective  distances.  And  when  the 
spaces  passed  over  by  the  two  bodies  in  their  orbits  are  as  the 
square  roots  of  their  respective  distances;  or  when  the  times 
passed  over  are  as  their  respective  distances;  then,  according 
to  the  notions  which  the  world  have  entertained  in  respect  to  the 
correspondence  between  the  deflection  of  a  planet  and  the  law 
of  falling  bodies,  the  deflection  or  fall  from  a  tangent  will  be 
equal  in  the  two  bodies ;  and  in  such  case  when  the  spaces 
passed  over  in  their  respective  orbits  are  as  1  to  7.74595,  the 
deflection  of  one  will  be  equal  to  that  of  the  other. 

But  however  plausible  such  hypothesis  may  appear,  it  is  nev- 
ertheless untrue  in  iact,  and  should  not  be  permitted  to  lead  into 
unbounded  errors. 

Recur  again  for  explanation  to  the  planets  A  and  C;  and 
upon  the  hypothesis  aforesaid,  if  C  had  passed  over  equal 
space  with  A,  (and  in  twice  the  time,)  its  deflection  would  be 
l-4th  that  of  A;  the  consequence  of  which  would  be,  when  C 
had  passed  over  twice  the  space  of  A,  its  deflection  would  be 
equal  to  that  of  A,  viz.,  when  the  spaces  passed  over  were  as  the 
square  roots  of  their  distances,  or  the  times  passed  over  were  as 
their  distances.  For  if  deflection  be  like  the  law  of  falling  bodies, 
then,  if  the  deflection  of  C,  in  equal  time,  be  l-16th  that  of  A, 
it  will  be  l-4th  in  equal  space,  and  in  quadruple  the  time  and 
double  the  space,  it  would  be  equal  to  that  of  A.  In  which  case 
when  C  has  passed  over  l-8th  of  its  orbit,  its  deflection  should  be 
just  equal  to  that  of  A  when  it  has  passed  over  l-4th  of  its  orbit, 
which  is  not  the  case;  and  hence  those  proportions  do  not  hold 
in  the  deflection  of  two  planets,  neither  in  the  expanse,  nor  in 
any  little  distance. 

Upon  the  foregoing  erroneous  hypothesis,  the  same  amount  of 
gravity  expended  upon  each  of  two  planets  of  a  system  (if  grav- 
ity be  inversely  as  the  square  of  the  distance)  would  give  the 
same  amount  of  deflection  in  each  ;  and  sixteen  times  the  amount 
of  gravity  or  rate  of  gravity,  would  give  twice  the  motion.  But 
if  gravity  be  inversely  as  the  distance,  then  the  same  amount 
expended  on  two  planets  of  a  system  will  give  the  same 


176  ON    THE    LAW    OF    GRAVITY. 

amount  of  deflection,  if  the  times  in  which  it  is  expended 
are  as  the  distances;  and  four  times  the  amount  of  gravity 
will  give  double  the  motion.  But  the  results  are  neither  of 
them  correct  as  to  deflection. 

Thus  in  respect  to  the  factitious  and  troublesome  element  of 
deflection,  whether  considered  in  reference  to  time  or  space,  it 
furnishes  but  little  useful  information  in  respect  to  the  laws  of 
force  and  motion ;  as  it  does  not  correspond  with  the  law  of 
falling  bodies,  as  has  been  too  implicitly  believed.  To  be  sure, 
from  an  attempt  to  get  the  rate  of  deflection  in  time,  we  obtain 
an  expression,  denoting  the  inverse  of  the  square  of  the  dis- 
tance; and  by  attempting  to  consider,  or  obtain,  the  rate  of 
deflection  in  space,  we  obtain  an  expression  denoting  the  in- 
verse of  the  distance ;  both  those  expressions,  we  perceive,  how- 
ever, are  wholly  founded  upon  assumption.  The  fact  is,  there 
has  been  a  mistake  in  supposing  that  the  progress  of  the  deflec- 
tion of  a  planet  from  a  tangent,  caused  by  the  progress  of  a 
planet  revolving  in  its  orbit,  was  identical  with,  or  corresponded 
with  the  fall  of  a  body  from  a  state  of  rest.  And,  but  for  this 
mistake,  the  world  might  have  been  saved  much  trouble.  For 
it  has  introduced  into  the  science  of  astronomy  that  kind  of 
scribe  rule,  by  which  things  must  be  cut  and  tried  until  they  fit; 
so  that  already  have  they  been  compelled  to  equate  some  thirty 
supposed  irregularities  of  the  moon,  said  to  be  the  legitimate 
offspring  of  physical  astronomy,  resulting  from  Clairaut's  recon- 
ciliation with  the  Newtonian  law  of  gravity ;  which  reconcilia- 
tion was  brought  about  by  more  complicated  ingenuity  than  most 
astronomers  and  mathematicians  are  willing  to  reinvestigate. 
For  having  adopted  an  hypothesis  in  respect  to  the  law  of 
gravity,  which  would  give  but  just  half  the  motion  to  the  apogee 
of  the  moon  which  it  actually  has;  and  being  compelled  to  sup- 
ply the  deficiency  of  gravity  from  extraneous  materials ;  the 
moon  has  in  consequence  been  thrown  into  more  perturbations 
than  ever  was  a  hunted  hare ;  while  those  who  have  essayed  to 
pursue,  have  been  more  sagacious  than  a  pack,  in  tracing  out 
and  exploring  her  devious  paths  caused  by  her  perturbations. 

I  must  confess,  that  I  never  have  been  reconciled  to  the  idea 
that  the  universe  was  all  an  error;  and  that  truth  was  only  to  be 
found  at  its  furthest  extremity,  after  groping  through  eternal 
darkness;  nor  am  I  well  satisfied  with  that  class  of  truths 
which  are  supposed  only  to  be  found  in  those  ultimate  ratios 
which  are  supposed  to  end  in  nothing;  but  I  prefer  that  ratio 
should  have,  at  least,  a  numerical  point  to  stand  on.  But  I  know 
not  of  a  more  notorious  instance  of  the  adoption  of  the  principle, 


ON    THE    LAW    OF    GRAVITY.  177 

that  the  path  of  eternal  error  is  the  road  to  truth,  than  that 
evinced  in  the  modern  researches  for  the  quadrature  of  the  cir- 
cle; in  which  not  a  single  step  that  is  taken  in  the  progress  (at 
least,  after  taking  up  the  line  of  march)  develops  or  discloses  a 
single  fact  material  to  the  issue.  Neither  the  area,  nor  any 
linear  measure  of  a  single  polygon  is  measured,  determined,  or 
known. 

The  summing  up  of  the  process  at  any  stage  of  the  proceed- 
ing comes  out  error.  Hence,  by  such  method  we  -have  no 
criterion  by  which  to  determine  whether  we  are  or  are  not  in 
the  right  path.  There  is  no  progression  or  succession  of  results, 
which  serves  constantly  to  confirm  us  in  the  correctness  of  our 
pursuit,  no  landmarks  by  the  way  to  cheer  us  onward,  and 
denote  the  progress  we  have  made,  no  pleasant  reminiscences, 
like  those  of  Homer's 

"  Wayfaring  man,  who  wanders  o'er 
In  thought,  the  length  of  road  he  trod  before." 

There  is  no  series  of  results  arising  from  the  application  of 
the  powers  and  roots  of  numbers,  denoting  demonstration,  which 
should  as  fully  and  conclusively  determine  the  final  result  as  any 
other  in  the  process. 

Our  whole  hope  and  consolation  in  the  process  seems  to  arise 
from  the  circumstance,  that  the  farther  we  carry  it,  if  we  sum  up 
our  accounts,  so  much  nearer  than  at  any  previous  step  we  find 
we  are  to  what  we  conceive  to  be  the  neighborhood  of  truth,  and 
we  hence  conclude  that  the  path  we  are  pursuing  leads  directly 
there ;  and  that  truth  will  be  found  at  the  terminus  of  our  route. 
But  this  is  supposition ;  it  is  not  conscious  truth ;  and  how  near 
the  terminus  of  the  process  the  truth  lies,  we  know  not.  And 
should  it  eventually  be  found  that  such  process  is  wholly  erro- 
neous, it  may  serve  to  weaken  the  faith  of  mankind  in  the  belief 
that  truth  is  only  to  be  sought  and  obtained  through  an  infinite 
series  of  errors. 

I  will  now  conclude  this  section  by  a  few  desultory  dictums 
and  remarks. 

The  mean  distance  of  a  planet  divided  by  the  period,  gives 
the  rate  of  mean  motion.  The  distance  divided  by  the  rate  of 
motion  gives  the  period.  Hence,  when  the  distance  is  1,  the 
rate  of  motion  is  equal  to  the  period.  When  the  distance  is  2, 
the  rate  of  motion  is  l-4th  of  the  period.  When  the  distance  is  3, 
the  rate  of  motion  is  l-9th  of  the  period,  and  so  on. 

If  gravity  be  inversely  as  the  distance,  the  period  divided  by 
the  distance  will  give  the  amount  of  gravity  expended  during 
the  period.  If  gravity  be  inversely  as  the  square  of  the  dis- 
tance, the  distance  divided  by  the  period  will  give  the  amount 
23 


178  ON    THE    LAW    OP    GRAVITY. 

of  gravity  expended  during  the  period.  Thus,  by  the  two  hy- 
potheses, the  amount  of  gravity  expended  during  the  period  of 
a  planet,  will  be  reciprocals  of  each  other ;  thus,  in  one  case,  the 
amount  of  gravity  of  C  during  its  period,  would  be  2, —  and  by 
the  other  hypothesis,  it  would  be  .5. 

If  the  spaces  passed  over  by  two  planets  of  a  system  are  as 
the  square  roots  of  the  distances  of  the  planets,  each  planet  will 
have  received  an  equal  amount  of  gravity  with  the  other.  So 
the  gravity  exerted  on  A  in  time  1,  is  equal  to  that  exerted  on  D 
in  time  60,  —  or  on  C  in  time  4. 

If  gravity  be  inversely  as  the  distance,  twice  the  motion  will 
balance  four  fold  the  gravity. 

If  gravity  be  inversely  as  the  square  of  the  distance,  twice  the 
motion  will  balance  sixteen  fold  the  gravity. 

Upon  the  hypothesis  that  gravity  is  inversely  as  the  square  of 
the  distance ;  and  also,  that  twice  the  motion  balances  quadru- 
ple the  force,  it  follows  as  a  corollary,  that  if  gravity  were  to  be 
increased  so  as  to  become  inversely  as  the  distance,  the  motions 
of  the  planets  throughout  a  system,  would  be  equal  one  with 
another,  viz.,  the  motion  of  C  would,  with  but  l-4th  the  rate  of 
gravity,  be  equal  to  that  of  A. 

But  the  hypothesis  that  gravity  is  inversely  as  the  square  of 
the  distance,  presents  in  its  consideration  strange  dilemmas, 
which  I  will  not  stop  here  to  recount.  Suffice  it  to  say  that  the 
advocates  of  such  hypothesis  claim  for  it,  that  twice  the  motion 
of  a  planet  balances  quadruple  the  force ;  when  nothing  is  more 
obvious  than  that  the  terms  of  the  hypothesis  require  sixteen  fold 
the  force  to  balance  twice  the  motion. 

But  notwithstanding  the  hypothesis,  that  gravity  is  inversely 
as  the  square  of  the  distance,  is  so  generally  accepted  and  re- 
ceived, astronomers  have  brought  strong  reasons  to  bear  against 
it ;  as  in  case  of  Jupiter  and  his  satellites,  when  compared  with 
the  earth  and  moon ;  for  notwithstanding  those  unphilosophical 
notions  in  respect  to  the  different  densities  of  the  planets,  they 
have  gone  upon  the  principle  that  twice  the  motion  requires 
quadruple  the  force ;  and  consequently,  that  gravity  is  inversely 
as  the  distance  ;  and  hence,  that  the  mean  motion  is  always  the 
square  root  of  the  mean  gravity. 

From  a  consideration  of  the  times  of  the  periods  of  Jupiter's 
satellites,  and  of  the  moon,  astronomers  have  estimated  the  attract- 
ing power  of  Jupiter  to  be  something  over  three  hundred  times 
as  great  as  that  of  the  earth.  We  will  suppose  then,  for  the  sake 
of  round  numbers,  that  the  attracting  power  of  the  earth  com- 
pared with  that  of  Jupiter,  is  as  1  to  324;  that  the  distance 
of  our  moon  is  1,  and  consequently  each  of  its  elements  1 ; 
and  that  Jupiter's  first  satellite  is  at  the  same  distance  from 


ON    THE    LAW    OF    GRAVITY.  179 

Jupiter's  centre,  that  the  moon  is  from  the  centre  of  the  earth. 
Now  in  such  case,  all  astronomical  calculations  have  gone  upon 
the  ground,  that  the  period  of  our  moon  would  be  eighteen  times 
the  length  of  the  period  of  Jupiter,  viz.,  that  the  rate  of  motion 
in  either  case  is  the  square  root  of  the  force ;  and  consequently, 
that  the  force  is  inversely  as  the  distance. 

Thus  the  world,  in  relation  to  the  law  of  gravity,  seem  to  have 
adopted  an  article  of  faith  which  they  have  not  been  able,  in  a 
single  instance,  to  practise  upon.  Nevertheless,  the  most  crude 
and  palpable  error  of  Sir  Isaac  Newton,  is  yet  held  more  sacred 
than  that  infinity  of  simple  truths  which  are  continually  urging 
us  to  reject  it. 

SECTION    SIXTH. 

The  attempt  of  astronomers,  to  account  for  all  the  smaller  mo- 
tions of  the  planets,  of  the  moon,  &c.,  —  as  the  aphelia,  the 
nodes,  the  precession  of  the  equinoxes,  &c.,  upon  the  principle 
of  disturbing  forces  caused  by  the  mutual  attraction  of  those 
bodies  upon  each  other,  —  upon  the  Newtonian  theory  of  gravity, 
and  upon  the  principle  upon  which  Clairaut  is  supposed  to 
have  accounted  for  the  motion  of  the  moon's  apogee,  goes  upon 
the  hypothesis  that  matter  is  alike  endued  with  an  innate  princi- 
ple of  gravity  or  attraction,  which  operates  on  all  other  matter 
with  a  force  or  power  of  attraction  in  an  inverse  ratio  to  the 
square  of  the  distance ;  and  hence  the  Newtonian  theory  of 
gravity,  has,  through  much  curious  contrivance,  been  converted 
into  a  system  of  mutual  disturbing  forces.  But  if  the  foundation 
upon  which  the  superstructure  is  raised  be  unsound,  the  super- 
structure, of  course,  must  fall. 

But  it  appears  to  me  that  much  of  the  reasoning  in  support 
of  both  the  Newtonian  theory  and  law  of  gravity,  as  applied  to 
what  they  are  pleased  to  call  physical  astronomy,  is  somewhat 
a  priori ;  as  having  discovered,  for  instance,  the  amount  of 
the  motions  of  the  aphelia  of  the  planets,  and  having  assigned 
the  cause  to  disturbing  forces  upon  each  other,  it  became  neces- 
sary then,  in  accordance  with  their  theory  and  law  of  gravity,  to 
assign  to  each  planet  its  proper  share  of  the  disturbing  force,  al- 
though it  has  not  been  as  easy  to  ascertain  the  disturbing  powers 
arising  from  what  has  been  conceived  to  be  the  different  densi- 
ties of  those  planets  which  have  no  satellites,  as  of  those  which 
have  ;  for  La  Lande  and  La  Grange  differ  widely  as  to  the  den- 
sity of  Venus,  —  La  Grange  assigning  its  density  in  proportion 
to  its  distance  from  the  sun,  as  he  does  also  that  of  all  the  other 
planets,  and  he  avows  that  he  saw  no  cause  for  assigning  their 
densities  otherwise. 


180  ON    THE    LAW    OF    GRAVITY. 

But  I  will  quote  a  little  from  Mr.  Vince,  who  has  given 
a  general  digest  of  this  theory,  with  its  reduction  and  equation  to 
a  supposed  coincidence  with  the  facts,  as  found  from  observa- 
tion. He  says :  "  Sir  Isaac  Newton  was  the  first  who  accounted 
for  this  motion  (the  precession  of  the  equinoxes.)  Having  prov- 
ed lhat  from  the  centrifugal  force  of  the  parts  of  the  earth  arising 
from  its  rotation,  the  equatorial  diameter  must  be  greater  than 
the  polar,  he  proceeded  to  show,  that  if  we  conceive  a  sphere  to 
be  inscribed  in  the  earth,  the  attraction  of  the  sun  and  moon  upon 
the  excess  of  the  quantity  of  matter  in  the  earth  above  that  of  the 
sphere,  will  cause  a  motion  in  the  plane  of  the  equator,  and  make 
the  points  where  it  intersects  the  ecliptic  go  backwards  upon  it. 
But  although  he  assigned  the  true  cause  of  the  precession,  it  is 
acknowledged  that  he  fell  into  an  error  in  his  investigations  of 
the  effect," 

This,  then,  is  substantially  the  basis  upon  which  they  proceed  : 
and  after  an  attempted  demonstration,  showing  that  from  the  ac- 
tion of  the  sun  alone,  the  annual  precession  would  amount  to 
21"  6/y/,  we  find  these  observations,  namely : 

"  This  would  be  the  precession  of  the  equinoxes  arising  from 
the  attraction  of  the  sun,  if  the  earth  were  solid,  of  a  uniform 
density,  and  the  ratio  of  the  diameter  as  229  to  230  ;  but  from 
what  follows,  if  the  greatest  nutation  of  the  earth's  axis  be  rightly 
ascertained,  the  precession  is  only  14"  5'";  which  difference  be- 
tween the  theory  and  what  is  deduced  from  observation,  must 
arise  either  from  the  fluidity  of  the  earth's  surface,  or  increase  of 
density  towards  the  centre,  or  the  ratio  of  the  diameters  being 
different  from  that  which  is  here  assumed  ;  or  probably  from  all 
the  causes  conjointly.  This  regression  of  the  equinoxes  (caused 
by  the  plane  of  the  equator  moving  backwards  upon  the  ecliptic) 
must  necessarily  cause  the  poles  of  the  earth  to  describe  circles 
about  the  poles  of  the  ecliptic,  in  a  direction  contrary  to  the  order 
of  the  signs,  setting  aside  the  effect  of  nutation." 

Thus  we  perceive  that  heaven  and  earth  may  fail,  but  not  one 
jot  or  tittle  of  the  Newtonian  law  or  theory  of  gravity  must  fail ; 
everything  may  be  wrong  but  his  theory  and  his  promulgated 
law  of  gravity ;  when,  in  case  of  all  his  disturbing  forces,  as  well 
as  in  respect  to  the  motion  of  the  moon's  apogee  (even  if  those 
motions  depended  upon  disturbing  forces)  his  law  of  grav- 
ity would  give  but  o^e-fourth  part  force  enough  to  produce  the 
effect. 

But  in  respect  to  the  precession  of  the  equinoxes,  we  find  an- 
other motion,  namely,  a  nutation  of  the  poles  of  the  earth,  which 
is  brought,  in  connection  with  the  precession,  to  thwart  and  per- 
plex the  theory,  and  cause  a  division  of  the  equation  designed 
for  the  precession. 


ON    THE    LAW    OF    GRAVITY.  181 

And  hence  we  find  Mr.  Vince,  under  the  head  of  "  Precession 
and  Nutation  arising  from  the  Action  of  the  Moon,"  giving  a  dis- 
sertation, mixed  with  commentaries  from  Dr.  Bradley,  upon  his 
observations  upon  the  star  Draconis  and  others,  in  which  refer- 
ence is  made  to  his  discovery  of  the  aberration  and  progressive 
motion  of  light,  and  in  which  it  is  said  that  Dr.  Bradley  was  the 
discoverer  of  the  inequalities  of  the  precession  of  the  equinoxes, 
and  the  nutation  of  the  earth's  axis,  arising  from  the  attraction  of 
the  moon  in  different  situations  of  its  nodes  ;  and  from  the  length 
to  which  the  dissertation  is  prolonged,  I  must  refer  the  reader 
mainly  to  the  article  itself. 

Nevertheless,  we  find,  toward  the  close  of  the  article,  this  pas- 
sage :  "  The  surprising  agreement,  therefore,  in  so  long  a  series 
of  observations,  taken  in  all  the  various  seasons  of  the  year,  as 
well  as  in  the  different  positions  of  the  moon's  nodes,  seems  to 
be  a  sufficient  proof  of  the  truth  both  of  this  hypothesis  and  also 
of  that  which  I  formerly  advanced,  relating  to  the  aberration  of 
light."  And  then  follows :  "  The  conclusion  derived  from  these 
observations  is,  that  the  gradual  diminution  of  the  obliquity  of 
the  ecliptic  to  the  equator,  does  not  arise  from  an  alteration  in  the 
position  of  the  earth's  axis,  but  from  some  alteration  in  the  eclip- 
tic itself;  because  the  stars  at  the  end  of  the  period  of  the  moon's 
node,  appeared  in  the  same  places  with  respect  to  the  equator,  as 
they  ought  to  have  done,  if  the  earth's  axis  had  retained  the  same 
inclination  to  an  invariable  plane,"  and  finally  the  whole  nutation 
is  fixed  at  19".  And  again,  Mr.  Vince  says  :  "  If  the  annual 
precession  arising  from  the  sun  be  taken  at  21"  6//;,  as  in  Article 
1022,  and  the  whole  precession  be  50",  then  the  part  arising  from 
the  action  of  the  moon  will  be  28"  45'",  hence  (Article  1034) 
the  density  of  the  moon  to  the  density  of  the  sun  is  as  28"  54'"  to 
21"  6"'  multiplied  by  .988,  equal  to  20"  8'",  which  ratio  does  not 
agree  either  with  the  proportion  deduced  from  the  tides,  or  with 
the  accurate  observations  of  Dr.  Bradley.  The  best  method  of 
settling  this  point  is  from  the  greatest  nutation."  And  finally  we 
find  the  point  settled,  and  equated,  so  as  to  make  that  part  of  the 
precession  arising  from  the  action  of  the  moon,  amount  to  35" 
39'",  and  that  part  arising  from  the  action  of  the  sun,  14"  36'". 

And  near  the  close  of  the  chapter,  we  find,  under  the  head  of 
objections,  which  appear  to  be  raised  for  the  purpose  of  answer- 
ing and  allaying  them,  the  following : 

"  Another  difficulty  that  may  arise  is  in  relation  to  our  having 
made  the  effect  of  the  sun's  force  to  be  about  one-third  part  less 
than  the  quantity  resulting  from  calculations  founded  on  hydro- 
statical  principles  and  the  hypothesis  of  an  uniform  density  of  all 
parts  of  the  earth.  But,  that  the  phenomenon  cannot  be  truly 


182  ON    THE    LAW    OF    GRAVITY. 

accounted  for  upon  this  hypothesis,  appears  from  the  concur- 
rence of  all  experiments  in  general ;  for,  whether  we  regard  the 
mensuration  of  the  degrees  of  the  earth,  the  accurate  observations 
of  Dr.  Bradley  on  the  proportions  and  times  of  the  tides,  the  case 
is  the  same,  and  requires  a  much  less  effect  from  the  action  of  the 
sun  than  results  from,  or  can  consist  with  the  said  hypothesis. 
But  if  the  density  of  the  earth,  instead  of  being  uniform,  is  sup- 
posed to  increase  from  the  surface  to  the  centre  (as  there  is  the 
greatest  reason  to  imagine  it  does,)  then  the  phenomenon  may 
be  easily  made  to  quadrate  with  the  principles  of  gravitation ; 
and  that  according  to  innumerable  suppositions  respecting  the 
law  whereby  the  density  may  be  conceived  to  increase." 

Now,  to  me,  this  is  a  most  extraordinary  commentary  upon 
the  Newtonian  theory  and  law  of  gravity,  as  it  must  be  to  any 
person  who  has  not  one  particle  of  faith  in  either. 

I  have  already  remarked,  that  what  are  called  the  disturbing 
forces  of  the  planets  upon  each  other,  are  by  astronomers  placed 
on  similar  grounds  to  the  supposed  disturbing  force  of  the  sun 
upon  the  moon  ;  and  hence  we  find  Mr.  Vince  saying,  in  treat- 
ing of  the  motion  of  the  moon's  apogee,  "  The  action  of  one 
planet  upon  another  to  disturb  its  motion  about  the  sun,  is  simi- 
lar to  the  action  of  the  sun  upon  the  moon,  to  disturb  its  motion 
about  the  earth.  The  general  equation  of  the  curve,  therefore, 
which  the  disturbed  planet  describes,  may  represent  that  which 
the  moon  describes  about  the  earth,  and  thence  the  irregularities 
of  the  moon's  motions  may  be  investigated." 

And  in  accordance  with  the  foregoing,  he  further  says,  "  These 
fluxional  equations  are  the  same  as  those  determined  by  Clair- 
aut,  Euler  and  Mayer,  in  their  treatises  upon  the  theory  of  the 
moon,  the  integration  of  which  is  a  problem  of  great  difficulty,"  &c. 
And  Mr.  Vince,  after  making  what  he  calls  a  near  approxima- 
tion to  the  ratio  between  the  mean  motion  of  the  moon's  apogee 
and  the  mean  motion  of  the  moon,  concludes  with  these  re- 
marks :  "  Hence  we  may  conclude  that  the  theory  of  gravity  is 
sufficient  to  give  the  true  motion  of  the  moon's  apogee." 

Now,  it  is  rather  a  sad  commentary  upon  the  intelligence 
and  independence  of  mind  in  enlightened  communities,  that  it 
should  be  the  boast  and  admiration  of  the  multitude,  who  are 
ready  to  burst  into  pa3ans  and  songs  of  triumph  at  these  suppos- 
ed achievements  in  the  science  of  astronomy,  by  those  who  are 
conceived  to  have  so  far  surpassed  the  ken  or  comprehension  of 
mortals,  that  very  few  persons  upon  the  face  of  the  globe  can  be 
expected  to  understand  the  cause  and  effect,  or  comprehend  the 
modus  operandi  by  which  a  knowledge  of  the  operations  have 
been  achieved  ;  and  if  there  is  too  much  arrogance  on  the  one 


ON    THE    LAW    OF    GRAVITY.  183 

part,  and  too  much  implicit  faith  on  the  part  of  the  multitude, 
such  arrogance  and  implicit  faith  have  doubtless,  in  a  measure, 
had  their  origin  in  a  kind  of  national  pride,  and  received  their 
impulse  from  those  who  were  desirous  to  make  the  author  of  the 
Principia  illustrious  ;  and  that,  too,  from  the  fact  or  circumstance 
that  not  more  than  two  or  three  of  his  contemporaries  were  able 
to  understand  it,  and  that  more  than  fifty  years  elapsed  before  its 
great  physical  truth  (the  law  of  gravity  according  to  Newton) 
was  finally  understood,  acknowledged  and  appreciated  ;  and  only 
then  by  the  ingenuity  of  M.  Clairaut,  by  means  of  a  problem 
which  Mr.  Vince  often  alleges  to  be  one  of  extreme  difficulty, 
and  which  the  most  astute  are  scarcely  able  to  investigate  and 
understand. 

It  is  said  that  Sir  Isaac  Newton,  in  extending  the  consequen- 
ces of  the  great  physical  truth  contained  in  the  Principia,  gave 
the  true  cause  of  the  precession  of  the  equinoxes,  namely, 
the  attractive  influence  of  the  sun  upon  the  surplus  matter  of 
the  earth  over  and  above  its  greatest  inscribed  sphere ;  al- 
though Mr.  Vince  says,  "  it  is  acknowledged  that  Sir  Isaac  fell 
into  an  error  in  his  investigation  of  the  effect ;"  and  that  the  cel- 
ebrated English  astronomer,  Dr.  Bradley,  discovered  a  nutation 
of  the  poles  of  the  earth  caused  by  the  attraction  of  the  moon  up- 
on the  surplus  matter  of  the  earth  over  and  above  its  greatest  in- 
scribed sphere ;  which  nutation,  or  motion  of  the  poles  of  the 
earth,  he  conceives  and  defines  to  be  performed  in  small  ellipti- 
cal orbits,  whose  diameters  are  about  18"  and  16"  about  the  ac- 
tual poles,  or  what  he  calls  the  mean  poles  of  the  earth,  or  per- 
haps around  what  may  properly  be  called  the  axis  of  the  heavens. 

This  discovery  of  Dr.  Bradley  certainly  caused  much  perplex- 
ity, for  a  time,  to  the  science  of  astronomy,  until  it  was  final- 
ly settled  (as  it  were  by  convention)  what  portion  of  the  influ- 
ence in  producing  the  precession  of  the  equinoxes  should  be 
assigned  to  the  moon,  and  what  to  the  sun;  for  as  no  division 
of  influences  would  agree  or  correspond  with  the  supposed  pro- 
portional densities  of  the  sun  and  moon,  as  determined  in  case 
of  the  tides;  nor  even  with  the  critical  observations  of  Dr.  Brad- 
ley ;  together  with  many  other  difficulties  that  were  found  to 
exist  when  tested  by  their  assumed  law  of  gravity ;  it  was  at 
length  found  necessary  to  cut  the  matter  short,  by  considering 
what  might  be  produced  by  the  nutation  caused  by  the  moon,  and 
assigning  the  remainder  to  the  sun.  And  hence  some  35"  39//y 
of  the  precession  was  assigned  to  the  influence  of  the  moon,  and 
14"  36' '  to  the  sun  ;  although  some  give  only  30"  to  the  moon, 
and  the  remainder  to  the  sun. 

Now  the  division  of  the  spoil  in  respect  to  the  small  motions 


184 


ON    THE    LAW    OF    GRAVITY. 


of  the  nutation  of  the  poles,  and  the  precession  of  the  equinoxes, 
which  has  been  portioned  out  to  the  sun  and  moon,  and  to 
certain  great  astronomers,  seems  to  have  arisen  from  Dr.  Bradley's 
critical  and  careful  observations  upon  the  apparent  motions 
of  the  fixed  stars  ;  and  in  particular  upon  the  star  Draconis ; 
from  which  phenomena  observed,  he  made  liberal  deductions 
as  to  causes  and  consequences  ;  as  a  nutation  of  the  poles 
of  the  earth,  and  a  division  of  the  influence  of  the  sun  and 
moon  in  producing  the  precession  of  the  equinoxes ;  as  also 
a  detection  of  the  progressive  motion  of  light  in  time,  its  ra- 
pidity &c.,  all  of  which  deductions  still  pervade  our  works  on 
philosophy  in  liberal  treatises,  embellished  by  diagrams,  and  what 
is  termed  the  higher  mathematics.  And  inasmuch  as  we  all 
draw  liberally  upon  the  Dr.'s  detailed  observations  in  making 
deductions  as  to  the  physical  causes  of  the  phenomena  observed, 
it  is  but  just  that  the  leading  features  in  his  account  of  them 
should  be  liberally  quoted,  with  the  view  that  all  may  compare 
the  phenomena  observed  with  the  various  deductions  and  hy- 
potheses drawn  from  them  as  their  physical  cause. 

In  respect  to  the  Dr.'s  hypothesis  of  the  progressive  motion  of 
light  in  time,  I  certainly  believe  it  to  be  a  very  vague  conjecture, 
which  should  never  have  been  incorporated  into  philosophy  as  a 
component  part,  clothed  with  all  the  dazzling  brilliancy  of  the 
higher  branches  of  mathematics,  and  consequently,  placed  where 
few  would  conceive  themselves  capable  of  investigating  the  truth 
or  error  of  the  hypothesis. 

If,  however,  the  detailed  observations  of  Dr.  Bradley,  on  the 
star  Draconis,  were  alone  to  be  considered,  perhaps  no  phenom- 
enon furnishes  more  conclusive  evidence  of  any  fact  connected 
with  it,  than  that  observed  by  Dr.  Bradley  in  favor  of  the  hy- 
pothesis that  light  possesses  no  progressive  motion.  But  in 
respect  to  the  nutation  of  the  poles,  it  is  believed  that  the  Dr. 
was  not  remiss  in  his  attempt  to  assign  the  physical  cause  ;  and 
it  is  also  believed,  that  he  is  the  only  one  who  ever  made  a  suc- 
cessful attempt  to  take  from  Sir  Isaac  Newton  anything  which 
he  had  found ;  nor  is  this  robbery  ;  it  is  but  a  division  of  the 
spoil,  —  leaving  to  the  lion  his  share,  viz:  his  theory  and  law  of 
gravity  ;  he  only  took  from  the  sun,  and  gave  to  the  moon ;  which 
was  certainly  in  accordance  with  the  Newtonian  theory.  But 
why,  after  the  Dr.  had  so  fully  assigned  what  he  and  everybody 
else  then  believed  to  be  the  physical  cause  of  the  phenomena,  it 
should  afterwards  have  been  alleged,  that  the  Dr.  was  the  first  to 
discover  by  observation,  the  apparent  change  in  the  declination  of 
the  stars,  (but  that  the  physical  causes  were  afterwards  explained 
by  D'Alembert  and  others,)  I  know  not,  unless  it  was,  that  they 


ON    THE    LAW    OF    GRAVITY.  185 

discovered  a  discrepancy  in  the  Dr.'s  deduction  of  the  progressive 
motion  of  light,  from  a  supposition,  that  the  apparent  declination 
of  the  stars  was  about  twice  the  amount  which  would  be 
assigned  to  them,  either  in  a  consideration  of  the  precession  of  the 
equinoxes,  or  in  a  nutation  of  the  poles;  for  surely  the  Dr.  in 
his  speculations  on  the  subject,  transports  us  through  a  whole 
period  of  the  moon's  nodes ;  at  the  end  of  which  he  again  finds 
things  as  they  were,  except  that  the  equinoxes  have  continued 
their  regressive  motion. 

There  has  been  no  disposition,  however,  in  any  of  these  as- 
tronomers, to  break  loose  from  the  Newtonian  theory  or  law  of 
gravity ;  and  inasmuch  as  Sir  Isaac  Newton,  in  extending  the 
consequences  of  the  great  physical  truth  of  the  Principia,  assign- 
ed the  phenomena  of  the  precession  of  the  equinoxes,  to  the 
direct  agency  of  gravity  upon  the  surplus  matter  about  the 
equator,  over  and  above  the  greatest  inscribed  sphere  of  the 
earth,  — of  course  the  same  has  never  been  questioned  or  doubted ; 
and  hence,  as  those  motions  of  the  earth  were  placed  directly 
under  the  Newtonian  theory  and  law  of  gravity,  and  inasmuch 
as  a  law  would  be  useless,  unless  it  were  obeyed,  it  is  said,  that 
D' Alembert  and  Laplace  have  succeeded  by  analysis  in  reducing 
all  these  intricate  phenomena  to  the  law  of  gravity  with  the 
most  complete  success  ;  and  such  is  generally  understood  to  be 
the  case.  And  inasmuch  as  the  labor  and  ingenuity  has  been 
so  vast  in  reducing,  adjusting  and  equating  them  to  the  Newto- 
nian law  of  gravity,  —  compared  with  what  it  would  have  been 
to  have  simply  accounted  for  them  upon  some  true  or  rational  hy- 
pothesis,—  it  might  seem  a  pity  to  attempt  to  disturb  the  decis- 
ion, even  upon  the  discovery  of  ample  and  satisfactory  evidence. 

Nevertheless,  I  am  bound  to  consider  that  those  who  have 
reconciled  these  phenomena  to  the  Newtonian  law,  have  no 
more  abided  by  the  two  maxims  laid  down  by  Newton,  than 
Sir  Isaac  himself  did ;  namely,  "  No  more  causes  are  to  be 
admitted  than  are  sufficient  to  explain  the  phenomena," — and 
that  of  "effects  of  the  same  kind,  the  same  causes  are  to  be 
assigned,  so  far  as  it  can  be  done." 

Now  aside  from  the  question  whether  the  rate  or  intensity 
of  gravity  be  inversely  as  the  distance,  or  inversely  as  the 
square  of  the  distance,  can  it  be  rationally  conceived,  or  even 
pretended,  that  the  position  and  motion  of  the  earth  requisite  to 
produce  the  precession  of  the  equinoxes,  or  the  nutation  of  the 
poles,  are  any  more  dependent  upon  the  direct  operation,  or  force 
of  gravity,  than  is  the  position  of  the  poles  or  equator  of  the 
earth,  in  respect  to  the  ecliptic,  or  the  rotatory  or  diurnal  motion 
of  the  earth  upon  its  axis  ? 
24 


186  ON    THE    LAW    OF    GRAVITY. 

And  if  not,  then  there  is  no  direct  physical  necessity,  that  the 
force  of  gravity  should  be  the  direct  or  immediate  cause,  and 
hence  it  should  not  be  assigned  as  the  immediate  cause,  unless 
such  shall  be  the  necessity  of  the  ease.  That  is,  if  the  phenom- 
ena may  take  place  or  exist  without  being  directly  or  immediately 
caused  by  gravity,  we  then  have  no  business  to  assign  gravity  as 
the  direct  cause.  We  then  may  draw  something  from  analogy 
in  sustaining  the  hypothesis,  that  the  earth  may  have  the  requi- 
site motions  for  producing  the  phenomena  under  consideration, 
without  resorting  to  the  immediate  agency  of  gravitation;  or 
without  resorting  to  the  ridiculous  idea  of  a  projection  of  the 
planets  in  tangents  to  their  orbits  in  the  outset  of  their  career. 

Now  the  nutation  of  the  poles  is  called,  and  conceived  to  be, 
Lunar  Nutation ;  and  it  is  said  that  Physical  Astronomy  has 
made  known  the  existence  of  another  nutation,  called  Solar 
Nutation ;  but  that  it  is  too  small  to  be  detected  by  observation. 
Thus  what  is  termed  Physical  Astronomy,  based  upon  the 
Newtonian  theory  and  law  of  gravity,  has  already  subjected  the 
moon  to  some  thirty  equations  for  the  correction  of  supposed 
perturbations  ;  and  the  earth  seems  destined  to  be  hunted  down 
in  much  the  same  way,  unless  the  world  should  happen  to  dis- 
cover that  the  attractive  influence  of  the  sun  at  his  distance,  is 
capable  of  producing  more  than  two  hundred  times  the  motion 
of  the  earth ;  that  the  attractive  influence  of  the  moon  is,  at  her 
distance  from  the  earth. 

Nevertheless,  Precession  and  Nutation,  are  actual  motion, 
and  not  merely  position,  as  the  obliquity  of  the  ecliptic  to 
the  equator,  or  to  the  poles  of  the  earth  would  be,  were  it  not 
for  the  precession  and  nutation ;  and  as  my  theory  is,  that  all 
the  motions  of  the  heavenly  bodies  are  caused  and  controlled 
(either  directly  or  indirectly)  by  force,  or  gravity ;  that  they  are 
the  effects  of  the  operation  of  force,  and  not  of  time;  hence, 
precession  and  nutation  will  fall  within  the  same  category  as 
that  of  any  other  motion  of  the  heavenly  bodies,  and  be  subject 
to  the  same  laws. 

SECTION    SEVENTH. 

It  is  not  my  design  to  give  a  theory  of  the  moon's  motions, 
but  to  endeavor  to  relieve  her  from  some  of  those  perturbations 
which  have  been  caused  by  the  attempts  of  astronomers  and 
mathematicians,  to  bring  their  theory  to  agree  with  observa- 
tion ;  and  for  authority  on  this  subject,  I  shall  mostly  refer  to 
Vince's  Astronomy,  vol.  2,  chap.  32nd,  in  which  the  theory  of  the 
moon,  as  now  understood,  seems  to  have  been  pretty  thoroughly 
elaborated. 


ON    THE    LAW    OF    GRAVITY.  187 

Mr.  Vince,  after  some  preliminary  remarks  on  the  subject, 
commences  thus  :  "As  the  attractive  force  varies  inversely  as  the 
square  of  the  distance,"  &c. 

Again,  Mr.  Vince  says,  "  Sir  Isaac  Newton,  having  found 
that  the  moon  was  retained  in  its  orbit  by  a  force,  which  at  dif- 
ferent distances  from  the  earth,  varied  inversely  as  the  square  of 
the  distance ;  and  concluding  from  analogy,  that  the  same  law 
of  attraction  might  take  place  between  all  the  bodies  in  the  sys- 
tem, applied  this  theory,  (called  the  theory  of  gravity,)  to  com- 
pute the  effect  of  the  sun's  attraction  upon  the  earth  and  moon, 
so  far  as  it  might  affect  the  relative  situation  of  the  latter  as  seen 
from  the  former ;  and  hence  he  discovered,  besides  the  irregular- 
ities before  observed,  other  small  irregularities  of  the  moon's 
motion,  whicli  also  were  found  to  agree  with  observations. 
From  these  and  other  applications  of  his  theory,  he  was  confirm- 
ed in  his  conjectures  concerning  the  principle  of  universal 
gravitation  ;  and  the  same  principle  having  since  been  further 
applied,  and  found  to  produce  conclusions  conformable  to  obser- 
vation, his  theory  of  gravity  is  now  firmly  established."  To 
which  is  appended  the  following  note,  as  farther  explanation  :  — 
"  Sir  Isaac  Newton  found  that  if  the  force  with  which  bodies 
fall  upon  the  earth's  surface  were  extended  to  the  moon,  and  to 
vary  inversely  as  the  square  of  the  distance  from  the  centre  of 
the  earth,  it  would  in  one  minute  draw  the  moon  through  a  space 
which  is  equal  to  the  versed  sine  of  the  arc  which  the  moon 
describes  in  a  minute.  He  concluded,  therefore,  that  the  moon 
was  retained  in  its  orbit  by  the  same  force  as  that  by  which 
bodies  are  attracted  upon  the  earth." 

And  the  truth  in  respect  to  the  foregoing  proposition  is  cer- 
tainly a  desideratum  ;  and  should  there  be  error,  its  detection 
may  serve  to  show  how  prone  we  are  to  receive  those  things  for 
truths,  at  the  hands  or  suggestions  of  others,  which  it  is  our  busi- 
ness and  our  privilege,  to  examine  for  ourselves ;  and  that  we 
ought  not  too  implicitly  to  rely  upon  the  truth  of  those  things, 
which  others  profess  to  have  found,  (however  renowned  for 
their  wisdom,)  lest  we  imbibe  error  in  place  of  some  great  lead- 
ing truth,  and  then  call  upon  heaven  and  earth  to  yield  their 
laws  to  sustain  and  corroborate  it. 

I  will  now,  in  short,  state  the  premises  upon  which  I  propose 
to  compare  my  result  in  respect  to  the  law  of  gravity  with  that 
of  Sir  Isaac  Newton's,  as  deduced  from  the  law  of  falling 
bodies  near  the  earth,  and  its  identity  with  that  of  gravity  which 
retains  the  moon  in  her  orbit. 

1.  For  the  purposes  of  calculation,  Newton,  Maclaurin  and 
others  have  gone  upon  the  ground  that  the  moon  is  sixty  times  as 


188  ON    THE    LAW    OF    GRAVITY. 

far  from  the  centre  of  the  earth  as  it  is  from  the  surface  of  the 
earth  to  its  centre. 

2.  By  the  law  of  falling  bodies,  it  is  found  that  a  body  falling 
from  a  state  of  rest  near  the  surface  of  the  earth  will  fall  in  one 
second  of  time  from  its  commencement,  about  sixteen  feet,  which 
may,  for  the  sake  of  ease  in  calculating,  be  called  one  rod,  or 
abstractly,  one  space ;  and  the  second  of   time  in  which  it  is 
falling,  may,  abstractly,  be  called  one  moment. 

3.  We  will  also  consent,  (for  the  present  purpose,)   that  Sir 
Isaac  Newton  found,  from  observation  and  calculation  upon  the 
orbit  of  the  moon,  and  her  motion  in  such  orbit,  that  the  versed 
sine  of  the  arc  of  the  moon's  orbit  described  in  60  seconds  of 
time  by  the  moon  while  revolving  in  her  orbit,  is  just  one  rod, 
or  one  space,  or  just  as  far  as  a  body  will  fall  from  a  state  of 
rest  near  the   surface  of  the  earth  in  one  second  of  time,   or 
that  the  moon,  at  the  distance  60  from  the  centre   of  the  earth, 
will  fall  just  as  far  in  60  seconds  or  moments  of  time,  as  a  body 
at  the  surface  of  the  earth,  or  at  l-60th   part  of  the  distance  of 
the  moon  from  the  centre   of  the  earth,  will  in  one  second  or 
moment  of  time. 

Now  these  are  the  premises  ;  and  the  question  to  be  deter- 
mined is,  whether,  to  produce  these  results,  the  force  of  gravity 
towards  the  earth  (or  centre  of  the  earth)  be  inversely  as  the 
square  of  the  distance,  as  taught  by  Sir  Isaac  Newton,  or 
simply  inversely  as  the  distance,  as  I  allege  it  to  be. 

These  are  the  premises  from  which  Sir  Isaac  Newton  and 
others  have  drawn  the  conclusion  that  the  force  of  gravity  varies 
inversely  as  the  square  of  the  distance  varies.  And  these  are 
the  premises  from  which  I  draw  the  conclusion  that  the  force  of 
gravity  varies  inversely  as  the  distance  varies. 

I  contend  that  Sir  Isaac  Newton  applied  the  inverse  of  the 
square  of  the  distance  to  the  intensity  or  rate  of  the  force  ap- 
plied ;  when  it  only  refers  to  the  amount  of  the  fall  in  given 
time ;  that  is,  in  definite  moments  of  time,  to  wrhich  it  does 
apply. 

For  the  purpose  of  making  a  few  additional  commentaries 
on  the  subject,  I  will  here  again  quote  Mr.  Maclaurin's  exposi- 
tion of  Sir  Isaac  Newton's  hypothesis,  in  respect  to  the  law  of 
gravity,  in  view  of  the  circumstance  that  he  was  a  mathematician 
of  celebrity,  and  also  a  personal  friend  of  Sir  Isaac  Newton.  He 
says, "  The  computation  may  be  made  after  this  manner :  the  mean 
distance  of  the  moon  from  the  earth  being  60  times  the  distance  of 
heavy  bodies  at  the  surface  of  the  earth  from  its  centre,  (a  body 
at  the  surface  of  the  earth  will  fall  15yV  Parisian  feet  in  a  sec- 
ond,) and  her  gravity  increasing  in  proportion  as  the  square  of 


ON    THE    LAW    OF    GRAVITY.  189 

the  distance  from  the  centre  of  the  earth  decreases,  her  gravity 
would  be  60  X  60  times  greater  near  the  surface  of  the  earth 
than  at  her  mean  distance,  and  therefore  would  carry  her  through 
60  X  60  X  15TV  Parisian  feet  in  a  minute  near  the  surface ;  but 
the  same  power  \vould  carry  her  through  60  x'  60  times  less 
space  in  a  second  than  in  a  minute,  by  what  has  been  often  ob- 
served of  the  descent  of  heavy  bodies  ;  'and  therefore  the  moon, 
in  a  second  of  time,  would  fall,  by  her  gravity,  near  the  sur- 
face of  the  earth,  15-^  Parisian  feet;  which  therefore  is  the 
same  with  the  gravity  of  terrestrial  bodies." 

Now  this  might  be  well  enough  for  an  article  of  faith,  if  a 
person  had  no  particular  regard  for  its  being  true.  But  the 
method  of  begging  the  question,  or  assuming  without  proof,  in 
matters  that  should  be  demonstrated,  may  not  always  make  the 
evidence  as  conclusive  as  it  may  be  specious. 

Mr.  Maclaurin  says,  "  a  body  near  the  surface  of  the  earth 
will  fall  ISyV  Parisian  feet  in  a  second ;  and  her  (the  moon's) 
gravity  increasing  in  proportion  as  the  square  of  the  distance 
from  the  centre  of  the  earth  decreases,  her  gravity  would  be  60 
X60  greater  near  the  surface  of  the  earth  than  at  her  mean  dis- 
tance ; "  (that  is,  in  case  that  gravity  increases  as  the  square  of  the 
distance  decreases,  the  gravity  of  the  moon  would  be  3600  times 
as  great  at  the  surface  of  the  earth  as  at,  her  mean  distance  ; 
which  would  be  a  fair  corollary  or  conclusion,  if  the  premises 
were  true;)  and  therefore"  (Mr.  Maclaurin  says)  would  carry 
her  through  60  X  60  X  l&iV  Parisian  feet  in  a  minute  (or  60 
seconds)  at  the  surface;"  —  thus  making  the  force  of  gravity,  or 
urging  force,  increase  with  the  acceleration,  during  the  fall." 

Mr.  Maclaurin  proceeds  thus  :  "  But  the  same  power  would 
carry  the  moon  60  X  60  times  less  space  in  a  second  than  in  a 
minute  by  what  has  been  often  observed  of  the  descent  of  heavy 
bodies ; "  (which  allegation  is  true,  as  found  from  observation  and 
experience,  whatever  be  the  law  of  gravity  ;)  "  and  therefore  "  (Mr. 
Maclaurin  says)  the  moon  in  a  second  of  time  would  fall  by  her 
gravity  near  the  surface  of  the  earth  15-&  Parisian  feet ;  which 
therefore  is  the  same  with  the  gravity  of  terrestrial  bodies." 

And  this  last  allegation  is  true,  for  the  simple  reason,  that  a 
body  falling  freely  from  a  state  of  rest,  near  the  surface  of  the 
earth,  will  fall  15^  Parisian  feet  in  one  second  of  time. 

But  why  the  word  therefore,  should  have  been  called  in  aid,  I 
know  not,  —  as  the  allegation,  that  the  moon  would  fall  15TV*Pa- 
risian  feet  in  one  second  of  time  near  the  surface  of  the  earth, 
certainly  is  not  a  corollary  or  consequent  flowing  from  his  first 
allegation,  viz  :  that  gravity  increases  inversely  as  the  square  of 
the  distance  decreases ;  as  his  .last  allegation  would  be  true  by 


190  ON    THE    LAW    OF    GRAVITY. 

any  law  of  gravity.  Neither  does  it  apply  to  his  second  allega- 
tion, as  that  also  would  be  true  by  any  law  of  gravity ;  for  it  is 
an  established  fact,  that  necessarily  applies  to,  and  is  controlled 
by  the  law  of  gravity,  whatever  that  law  may  be. 

The  whole  of  Mr.  Maclaurin's  or  Sir  Isaac  Newton's  sup- 
posed demonstration,  or  explanation  of  the  law  of  gravity,  as 
conceived  to  have  been  deduced  from  the  law  of  falling  bodies, 
then  vanishes  in  thin  air.  They  have  only  seduced  themselves 
by  a  shadow  of  demonstration,  where  no  substance  exists. 

They  have  laid  no  premises  from  which  conclusions  are 
drawn,  save  the  assumed  or  unproved  and  erroneous  allegation, 
that  the  force  of  gravity  is  inversely  as  the  square  of  the  dis- 
tance;  and  yet,  upon  those  assumed  premises,  rests  most  of 
the  fabric  of  modern  astronomy. 

But  what  the  world  would  be  glad  to  know,  is,  whether  the 
force  of  gravity  decreases  as  the  distance  increases,  or  as  the 
square  of  the  distance  increases  ;  and  upon  this  point,  Mr.  Mac- 
laurin  has  given  us  no  proof  or  satisfaction  whatever. 

And  I  will  here  protest  against  this  hap-hazard  manner  of 
dispensing  science  to  the  world,  —  either  in  respect  to  the  quadra- 
ture of  the  circle,  or  the  law  of  gravitation,  —  wherein  no  premises 
are  laid  from  which  logical  deductions  can  be  drawn,  although 
conclusions  are  made  and  sent  forth  to  the  world,  which  too  often 
stifle  further  inquiry,  from  an  implicit  faith  in  those  who  have 
assumed  to  make  them.  But  I  deny  to  the  Pope  himself,  the 
right  thus  to  sell  indulgences  to  the  world. 

Mr.  Maclauriri  has,  however,  well  said,  "  that  the  moon  would 
fall  through  3600  times  less  space  in  one  second,  than  in  one 
minute,  (or  60  seconds,)  by  what  has  often  been  observed  of  the 
descent  of  falling  bodies  ;  "  —  thus  consenting  to  the  law  of  fall- 
ing bodies  as  it  is  ;  and  because  such  is  the  law  of  falling  bodies 
the  force  of  gravity  varies  inversely  as  the  distance  varies. 

A  falling  body  then,  urged  by  a  constant  and  uniform  force 
through  60  seconds  of  time,  from  a  state  of  rest,  will  fall  3600 
times  as  far  as  in  the  first  second  of  its  fall.  Hence  if  a  body 
at  the  surface  of  the  earth  fall  over  one  rod,  or  198  inches,  in  the 
first  moment  (or  second)  of  its  fall,  the  moon  will  fall  over  a  lit- 
tle more  than  l-20th  of  an  inch  in  the  first  moment  or  second  of 
its  fall. 

Thus,  if  both  bodies  commence  their  fall  at  the  same  time, 
that  at  the  surface  of  the  earth  will  fall  3600  times  as  fast,  at  any 
given  instant,  as  that  at  the  distance  of  the  moon  will.  So  if 
the  time  of  the  fall  of  the  moon  be  60  seconds,  it  will  fall  119 
times  as  far  in  the  last  second  or  moment,  as  in  the  first,  and 
with  the  same  amount  of  force  expended ;  and  the  same  ratio 
will  hold  as  to  the  fall  of  the  body  near  the  earth. 


ON    THE    LAW    OF    GRAVITY.  191 

Thus  we  see  that  time  only  compensates  for  the  lack  of  the 
intensity  or  rate  of  force,  just  in  accordance  with  the  laws  of 
dynamics,  from  the  days  of  Archimedes  to  the  present ;  but  does 
not  supply  the  place  of  force,  or  compensate  for  it,  in  any  other 
way  than  that  of  a  lack  or  want  of  intensity. 

Thus  Archimedes  could  move  the  world  with  his  lever,  if  he 
had  a  proper  fulcrum  and  a  place  to  stand  on  ;  and  the  allega- 
tion is  true  in  theory.  But  we  learn  from  his  notions  of  the 
operations  of  time,  force,  and  motion,  that  he  would  expect  no 
assistance  from  time ;  otherwise  than  that,  if  he  would  take  time 
enough  for  the  accomplishment  of  his  enterprize,  it  would  com- 
pensate for  his  weakness. 

This  principle  in  dynamics  comes  also  under  the  most  com- 
mon observation  ;  as  in  the  common  use  of  the  lever  in  raising 
weights  by  a  windlass,  whether  with  a  horizontal  or  upright 
shaft ;  as  well  as  in  the  use  of  all  the  mechanical  powers,  in  which 
we  make  use  of  time  to  compensate  for  the  lack  or  want  of  the 
intensity  of  force. 

Nevertheless,  the  same  amount  of  force  is  necessary  to  accom- 
plish the  object,  whether  the  time  be  longer  or  shorter ;  for  time 
does  not  do  the  work,  but  only  furnishes  the  time,  and  hence 
does  not  compensate  otherwise  than  for  a  lack  of  power  to  do  it 
sooner. 

Hence  there  will  be  just  as  much  force  of  gravity  expended 
in  60  seconds  at  the  distance  of  the  moon,  as  in  one  second  at 
the  surface  of  the  earth  ;  and  in  either  of  the  60  seconds  at  the 
distance  of  the  moon,  there  will  bel-60th  part  as  much  force  ex- 
pended, as  in  one  second  at  the  earth's  surface. 

But  if  the  force  of  gravity  be  inversely  as  the  square  of  the 
distance,  I  believe  all  will  readily  agree,  that  the  whole  amount 
of  force  expended  on  the  moon  in  sixty  seconds  of  time,  will  be 
but  the  l-60th  part  as  much  as  that  expended  on  the  body  near 
the  earth  in  one  second  of  time ;  in  which  case,  at  the  distance 
of  the  moon,  time  must  actually  supply  the  place  of  59-60ths  of 
the  force  requisite  to  urge  it  over  the  given  space,  unless  we 
adopt  another  hypothesis,  equally  though  no  more  absurd,  name- 
ly, by  supposing  the  force  to  increase  as  the  motion  accelerates, 
or  as  the  odd  numbers  in  their  order ;  in  which  case,  119 
times  the  force  that  is  expended  in  the  first  would  be  expended 
in  the  last  second  of  time. 

For  no  one  will  contend,  if  the  force  of  gravity  be  inversely 
as  the  square  of  the  distance,  that  more  than  l-60th  part  of  the 
force  is  expended  on  the  moon  that  is  expended  on  the  body 
near  the  earth  while  passing  over  equal  space. 

But  if  the  force  of  gravity  be  inversely  as  the  distance,  the 


192  ON    THE    LAW    OF    GRAVITY. 

same  amount  of  gravity  or  force  will  be  expended  on  the  moon 
while  falling  over  a  given  space,  as  will  be  expended  on  a  body 
near  the  earth  while  falling  over  an  equal  space,  but  in  l-60th 
part  of  the  time  ;  in  which  case  the  lack or  want  of  intensity  of 
force  at  the  distance  of  the  moon,  would  be  compensated  for  by 
time,  in  accordance  with  all  the  laws  of  dynamics  with  which 
we  are  acquainted. 

Hence  we  arrive  at  this  important  fact  in  the  law  of  falling 
bodies,  namely,  if  we  conceive  an  individual  body  to  fall  over  a 
given  space,  and  that  the  time  of  its  fall  be  divided  into  any 
number  of  equal  moments,  we  must  also  conceive  the  amount 
of  force  applied  in  each  moment,  to  be  equal  one  with  another ; 
then,  in  the  first  moment,  if  the  amount  of  space  fallen  over  be 
called  1,  the  amount  fallen  over  in  the  second  moment  will  be  3, 
only  the  like  amount  of  time  or  force  being  expended  as  in 
the  first  moment ;  in  the  third  moment,  the  spaces  will  be  5,  and 
so  on,  by  the  odd  numbers ;  in  which  case,  at  the  end  of  any 
given  moment,  the  whole  number  of  spaces  fallen  over  will  be 
as  well  the  square  of  the  several  amounts  of  force  expended  in 
all  the  several  moments,  as  of  the  sum  of  the  moments  them- 
selves. 

But  if  we  compare  two  bodies  commencing  their  fall  at  the 
same  time  towards  the  gravitating  or  attracting  power,  at  different 
distances  from  such  power,  as  in  case  of  the  moon  and  a  body 
near  the  surface  of  the  earth,  namely,  one  body  at  the  distance  1, 
and  another  body  at  the  distance  60,  for  example,  we  shall 
have  the  following  results,  namely :  It  is  manifest  that  if 
we  conceive  the  two  bodies,  one  near  the  surface  of  the  earth, 
at  the  distance  1,  and  the  moon  at  the  distance  60,  to  have  fallen 
over  an  equal  space,  namely,  one  rod  for  example  ;  that  near  the 
earth  in  one  second,  and  the  moon  in  60  seconds  of  time ;  then, 
taking  the  seconds  as  the  time,  the  times  of  each,  in  falling  over 
equal  space,  will  be  as  their  respective  distances  from  the  centre 
of  gravity ;  while  the  force  employed  on  each  will  be  as  the 
amount  of  space  fallen  over.  Hence,  in  either  case,  the  amount 
of  space  fallen  over  or  described,  will  be  the  square  of  the  whole 
amount  of  force  employed  in  the  fall ;  while  the  amount  of  force 
employed  upon  the  moon  during  its  fall,  is  but  l-60th  part  of 
the  time  of  its  fall. 

Hence,  the  Newtonian  law  of  gravity  essays  to  convert  time 
into  actual  force,  in  lieu  of  permitting  it  to  perform  its  legitimate 
office  of  compensating  for  want  of  intensity  of  force,  which  has 
been  well  known  to  be  its  legitimate  office  for  thousands  of  years. 

Thus  have  I  endeavored  fairly  to  present  the  evidence,  the  ma- 
terials, the  elements,  from  which  Sir  Isaac  Newton  deduced  his 


ON    THE    LAW    OF    GRAVITY.  193 

conclusion  that  the  force  of  gravity  is  inversely  as  the  square  of 
the  distance,  and  from  which  materials  or  elements  I  deduce  my 
conclusion  that  the  force  of  gravity  is  inversely  as  the  distance. 

And  hence,  as  a  corollary  or  consequent  of  my  determination, 
if  the  mean  motion  of  a  planet  (whether  revolving  in  a  centric  or 
eccentric  orbit)  were  to  be  doubled  in  order  to  retain  it  in  its  or- 
bit, the  mean  force  would  only  have  to  be  quadrupled,  or  in- 
creased four-fold,  in  lieu  of  being  increased  sixteen-fold,  as  would 
be  requisite  by  the  Newtonian  law  of  gravity  ;  and  consequently, 
the  motion  of  the  moon,  whether  in  its  simple  revolution 
about  the  earth,  or  in  the  motion  of  its  apogee,  (as  well  as 
the  motion  of  each  heavenly  body,)  would  be  entitled  to  just 
four  times  as  much  instant  or  constant  force  by  my  determina- 
tion of  the  law  of  gravity,  as  by  that  of  Sir  Isaac  Newton,  as  will 
be  readily  perceived  by  recurring  to  a  system  of  planets  which 
revolve  about  one  and  the  same  central  power,  as  in  case  of  the 
planets  about  the  sun. 

And  now,  —  notwithstanding  my  belief  is  that  I  have,  in  the 
most  direct  and  conclusive  manner,  confuted  the  Newtonian  deter- 
mination in  respect  to  the  law  of  gravity,  by  showing  that  he  as- 
signed the  square  of  the  distance  to  the  intensity  of  the  force, 
when  it  should  have  been  assigned  to  the  amount  of  the  fall,  (or 
to  what  they  have  seen  proper  to  call  the  amount  of  the  deflec- 
tion of  the  planet  from  a  tangent  to  its  orbit,)  or  the  result  of  the 
force  applied,  —  I  will,  byway  of  reductioad  absurdam,  present 
some  few  of  the  difficulties  and  perplexities  under  which  the  phi- 
losophers, the  astronomers,  and  the  mathematicians  of  Europe 
for  a  long  time  labored,  in  their  endeavors  to  reconcile  the  New- 
tonian law  of  gravity  with  observed  phenomena,  and  the  facts 
that  were  found  from  observation  to  exist ;  and  in  which  what 
they  supposed  themselves  required  to  adopt  as  a  first  principle, 
namely,  the  Newtonian  law  of  gravity,  led  to  so  much  confusion 
and  mystery,  as  to  cause  them  for  a  long  time  to  conspire  and 
rebel  against  the  law  so  promulgated;  until  at  length,  in  despair 
of  extricating  themselves  from  its  unjust  requirements,  they  as- 
sented to  and  ratified  the  treaty  so  ingeniously  devised  by  the 
great  French  mathematician  Clairaut. 

And  here  I  will  render  the  only  rational  excuse  which  I  can 
devise  in  justification  of  their  final  acquiescence  in,  and  acknowl- 
edgment of  the  Newtonian  law  ;  namely,  that  they  could  not  con- 
ceive an  error  to  have  been  made  in-  respect  to  the  well  known 
law  of  falling  bodies;  and  hence,  they  even  considered  it  use- 
less, as  well  as  derogatory  to  Newton,  to  investigate  what  he 
had  determined  as  being  the  result  of  so  simple  and  easy  a  cal- 
culation. 

25 


194  ON    THE    LAW    OF    GRAVITY. 

But  this  is  a  subject  which  I  propose  to  press  upon  the  world; 
—  to  ask  the  institutions  of  my  own,  and  of  foreign  countries, 
to  reinvestigate  ;  and  that  I  shall,  eventually,  have  supporters,  who 
will  dare  to  risk  their  reputation  by  venturing  to  think  for  them- 
selves, I  have  no  doubt. 

But  to  refer  to  some  of  the  difficulties  under  which  the  math- 
ematicians and  astronomers  of  Europe  have  labored,  in  respect 
to  the  law  of  gravity,  as  promulgated  by  Newton,  I  quote  from 
Mr.  Vince,  who  though  a  very  honest  recorder  of  scientific  dis- 
coveries, is  nevertheless,  quite  partial  to  Sir  Isaac  Newton. 

Mr.  Vince,  in  the  introductory  remarks  to  his  treatise  upon  the 
Theory  of  the  Moon,  says,  "  To  enter  into  a  computation  of  all 
the  effects  of  the  disturbing  force  of  the  sun  upon  the  moon, 
and  their  nature,  would  require  the  investigation  of  the  nature 
of  a  curve  described  by  a  body  attracted  to  two   points,  called 
the  Problem  of  three  Bodies  ;"  which  problem,  he  suggests,  has 
been  solved  by  but  very  few  mathematicians,  amongst  whom 
are  Clairaut  and  some  few  others.     And  it  appears,  that  for  more 
than  half  a  century  from  and  after  the  time  that  Sir  Isaac  New- 
ton  promulgated   his  theory  and   law  of  gravity,  the   learned 
world    essayed   to  apply  them  to  the  motions  of   the    moon. 
Mathematicians    were    exceedingly  perplexed   with  Sir    Isaac 
Newton's  law  of  gravity ;  for  Mr.  Vince  proceeds  thus :    "  In 
the   year   1747,    M.    Clairaut,  in   a  memoir    read   before    the 
Academy  of  Sciences,  in  Paris,  made  an  objection  to  this  law, 
upon  this  ground;   that  it  would  not  account  for  the  motion  of 
the  moon's  apogee,  it  giving,  according  to  his  calculations,  that 
motion  only  one  half  of  what  it  was  found  to  be  by  observation  ; 
and  he  concluded,  that  it  was  necessary  to  change   this  law, 
by  adding  something  to  correct  it.     He  however  soon  after  dis- 
covered his  mistake,  and  was  the  first  who  gave  a  complete  the- 
ory of  the  moon,  in  which  he  showed  that  Sir  Isaac  Newton's 
law  of  gravity  would  not  only  account  for  the  motion  of  the 
moon's  apogee,  but  also  for  all  the  irregularities  of  the  moon," 
and  Mr.  Vince  proceeds  to  say,  that,  "  M.  Euler  has  done  great 
justice  to  M.  Clairaut  upon  his  solution  of  this  important  prob- 
lem, in  a  letter  to  Rev.  Casper  Witstien  ;  in  which  he  observes, 
that,  <  This  question  is  of  the  last  importance  ;  and  I  must  own, 
that  till  now,  I  always  believed   that  this  theory  did  not  agree 
with  the  motion  of  the  apogee  of  the  moon.'     M.   Clairaut  was 
of  the  same  opinion ;  but   he  has  publicly  retracted  it,  by  de- 
claring that  the  motion  of  the  apogee  is  not  contrary  to  the  New- 
tonian theory.     Upon   this   occasion,   I  have   renewed    my  in- 
quiries on  this  affair ;  and  after  most  tedious  calculations,  I  have 


ON    THE    LAW    OF    GRAVITY.  195 

at  length  found  to  my  satisfaction,  that  M.  Clairaut  was  in  the 
right,  and  that  this  theory  is  entirely  sufficient  to  explain  the 
motion  of  the  apogee  of  the  moon.  As  this  inquiry  is  of  the 
greatest  difficulty,  and  those  who  hitherto  pretended  to  have 
proved  this  nice  agreement  of  the  theory  with  the  truth,  have 
been  much  deceived,  it  is  to  M.  Clairaut  that  we  are  obliged 
for  this  important  discovery,  which  gives  quite  a  new  lustre  to 
the  theory  of  the  great  Newton  ;  and  it  is  but  now,  that  we  can 
expect  good  astronomical  tables  of  the  moon." 

And  in  accordance  with  the  foregoing  suggestions  of  M.  Eu- 
ler,  we  find  from  Mr.  Vince  that  various  had  been  the  attempts 
to  reconcile  the  Newtonian  law  of  gravity  with  fact  and  obser- 
vation in  respect  to  the  motion  of  the  moon's  apogee ;  for  Mr. 
Vince  says  that  "  M.  Walmsley,  in  his  theory  of  the  Motion  of 
the  Apsides,  has  computed  the  mean  motion  of  the  moon's  ap- 
ogee, and  his  conclusion  agrees  very  well  with  observation ;  but 
his  principles  are  altogether  wrong ;  for  he  has  entirely  omitted 
that  part  of  the  force  which  acts  in  a  direction  perpendicular  to 
the  radius,  which,  as  we  have  shown,  produces  just  one  half  of 
the  motion ;  he  also  assumes  the  mean  disturbing  force  in  the 
direction  of  the  radius  as  acting  constantly,  instead  of  the  real 
disturbing  force ;  and  he  also  wrongly  computed  the  periodic 
time  of  the  moon ;  it  was  by  accident  therefore  that  he  obtained 
the  mean  motion  ;  in  respect  to  the  true  motion  his  conclusions 
are  erroneous  ;  he  says  Mr.  Machin  has  not  given  us  his  pro- 
cess ;  we  cannot  therefore  say  how  far  it  was  just.  He  says 
also,  in  the  Phil.  Trans,  of  1751,  that  Mr.  P.  Murdock  has  given  a 
method  of  computing  the  mean  motion  of  the  moon's  apogee, 
by  first  considering  only  that  part  of  the  disturbing  force  which 
acts  in  the  direction  of  the  radius  ;  and  then  instead  of  supposing 
the  earth  to  be  at  rest,  by  conceiving  the  earth  and  moon  to  re- 
volve about  their  common  centre  of  gravity,  he  imputes  about 
one  half  of  the  motion  to  that  cause,  and  thence  deduced  a  con- 
clusion agreeing  with  observation. 

"  What  we  have  already  observed  (Art.  861)  is  sufficient  to 
show,  that  no  part  of  the  effect  can  arise  from  the  latter  circum- 
stance ;  and  he  has  also  (as  we  have  already  shown)  omitted  a 
cause  which  produces  about  one  half  the  motion ;  by  two  mis- 
takes he  has  therefore  fallen  upon  a  true  conclusion."  And  Mr. 
Vince,  in  conclusion,  makes  these  remarks :  "but  in  whatever 
manner  this  subject  is  treated,  some  corrections  are  applied  from 
observations,  in  order  to  render  the  equations  more  perfect ; 
not  that  the  principle  of  attraction  is  insufficient  to  furnish 
conclusions  which  shall  agree  with  observation,"  &c.  And 
again  :  "  Thus  we  have  given  the  reader  all  the  satisfaction  we 


196  ON    THE    LAW    OF    GRAVITY, 

are  able  upon  this  difficult  subject,  without  entering  into  a  direct 
solution  of  the  problem,  which  requires  the  integration  of  a 
fluxional  equation  of  the  second  order,  and  this  can  be  done 
only  by  an  approximation  of  a  very  intricate  nature,  and  of 
great  labor." 

The  reader  or  inquirer  who  will  take  the  trouble  to  examine 
the  moon's  motions,  and  especially  the  motion  of  its  apogee,  as 
contained  in  Mr.  Vince's  Astronomy,  will  find  that  in  all  cases, 
the  little  motions  and  inequalities  of  motion,  are,  in  whole  or 
in  part,  assigned  to  the  disturbing  force  of  the  sun  upon  the 
moon  ;  and  after  much  laboring  of  the  author  on  the  subject,  we 
find  this  allegation,  namely :  "  If  the  force  perpendicular  to  the 
radius  vector  be  neglected,  the  motion  of  the  apsides  comes  out 
one  half  of  what  is  here  determined,  and  this  we  know  from 
other  principles.  This,  therefore,  tends  to  confirm  the  legality  of 
the  method  here  employed," — which  allegation,  after  all,  I  think, 
must  be  taken  for  granted,  rather  than  as  being  proved ;  and  in 
respect  to  the  whole  matter,  I  have  come  to  the  following 
doubts : 

I  think  it  doubtful,  whether  the  student  or  inquirer,  on  a  care- 
ful examination  of  the  whole  subject,  will  readily  agree  with  the 
modus  operandi  by  which  the  observed  phenomena  of  nature  have 
been  supposed  to  be  reconciled  with  the  Newtonian  law. 

Whether  he  will  find  the  ratio  between  the  force  applied  to 
the  moon,  and  the  moon's  motion  in  its  orbit,  clearly  and  dis- 
tinctly assigned  or  stated. 

Whether  he  will  clearly  discover  how  they  have  obtained  the 
force  which  acts  perpendicular  to  the  radius  vector,  which  is  said 
to  give  just  half  the  motion  of  the  apogee. 

Whether  he  will  be  able  to  discover  by  what  law  of  gravity, 
the  distance  may  be  assumed  at  unity,  and  its  mean  force  of 
gravity  at  less  than  unity ;  or  in  other  words,  whether  according  to 
Mr.  Vince,  the  mean  force  of  the  moon  towards  the  earth  is 
equal  to  the  natural  gravity  of  the  moon  towards  the  earth, 
diminished  by  the  disturbing  force  of  the  sun  upon  the  moon  in 
the  line  of  the  radius  vector, —  together  with  many  other  matters 
equally  dubious,  which  the  inquirer  is  required  to  pass  through, 
in  order  that  he  may  be  brought  to  render  assent  to  their  results, 
or  rather  to  their  conclusions. 

And  finally,  I  am  doubtful  whether  the  inquirer  will  readily 
discover,  that  even  Sir  Isaac  Newton  understood  the  subject  in 
the  same  way  and  manner  in  which  it  is  now  supposed  to  be 
understood,  notwithstanding  Mr.  Vince's  supposition  of  what 
Newton  must  have  meant  by  some  very  obcure  passages  in 
his  Principia  ;  and  more  especially,  when  it  is  acknowledged  by 


ON    THE    LAW    OF    GRAVITY.  197 

Mr.  Vince,  that  Sir  Isaac  Newton  left  out  the  force  which  is  now 
said  to  act  perpendicularly  to  the  radius  vector;  and  which  is  said 
to  give  just  one  half  of  the  motion  to  the  apogee. 

Nevertheless,  Walmsley,  Murdock,  and  others,  have  been 
somewhat  cavalierly  treated,  for  their  attempts  to  make  the  New- 
tonian law  of  gravity,  account  for  the  phenomena  of  nature, 
which,  manifestly,  are  the  result  of  some  kind  of  law  of  gravity  ; 
notwithstanding,  they  made  their  conclusions  agree  very  well 
with  observation.  And  I  must  say,  that  I  think  the  treatment 
they  received  from  the  inquisition — from  those  who  would  settle 
the  controversy  by  establishing  articles  of  implicit  faith,  however 
my  terious  —  was  somewhat  harsh  and  arbitrary. 

Walmsley  and  others,  made  use  of  all  the  force  that  was  al- 
lowed them  by  the  Newtonian  law  of  gravity ;  and  as  to  the 
remainder  that  was  found  necessary  for  the  purposes  intended, 
each  and  every  one,  (not  even  excepting  Clairaut  himself,) 
picked  it  up  from  such  sources  as  were  in  his  view  most  avail- 
able ;  and  I  have  yet  to  learn  why  others,  (as  well  as  Clairaut,) 
had  not  the  right  to  call  to  their  aid,  for  the  purpose  of  triumph- 
ing over  great  difficulties,  such  auxiliary  forces  as  they  could 
best  command. 

But  wisdom  would  have  been  profitable  to  direct ;  and  had  those 
great  mathematicians,  in  lieu  of  confiding  in  their  high-priest, 
just  spent  an  hour  or  two  in  examining,  to  see  from  whence  he 
derived  his  authority,  they  would  not  have  been  thrown  into  all 
these  inexplicable  dilemmas.  They  would  have  found  just  force 
enough  to  account  for  all  the  observed  phenomena,  without 
resorting  to  anything  factitious  or  unnatural. 

Nevertheless,  however  gloomy  the  stubborn  fact  may  be,  if  Sir 
Isaac  Newton's  law  of  gravity  be  erroneous,  it  wholly  destroys 
his  whole  system  of  philosophy ;  and  in  such  case,  his  iheory  of 
universal  gravity,  (as  by  him  promulgated,)  ceases  to  exist ;  and 
the  universe  will  again  be  governed  by  immutable  and  controll- 
ing laws,  in  lieu  of  depending  on  fortuity  or  chance. 


CHAPTER    III. 


Of  the  Proper  Elements  from  which  to  determine  the  Laws  of 
Force  and  Motion,  incident  to  the  Heavenly  Bodies  in  their 
Eternal  Rounds. 

SECTION    FIRST. 

HAVING  attempted  to  show  that  what  astronomers  have  called 
the  deflection  of  a  planet,  or  the  deflection  of  a  planet  from  a 
tangent  to  its  orbit,  and  which  they  have  made  use  of  as  an  ele- 
ment, in  their  endeavors  to  account  for  the  agreement  between 
the  force  and  motion  by  which  a  planet  revolves,  is,  at  least,  but 
a  factitious  element,  and  not  applicable  to  the  laws  of  force  and 
motion,  as  operating  in  the  curve  of  an  orbit,  nor  identical  in  its 
mathematical  construction  with  that  of  the  law  of  the  rectilinear 
fall  of  a  body  through  space,  —  I  will  now  proceed  to  treat  of  the 
proper  elements  of  the  planets  from  which  to  deduce  the  modus 
operandi  by  which  they  are  retained  and  controlled  in  their 
orbits  while  performing  their  eternal  rounds, — which  elements 
I  will  call  by  the  appellations  of  Time,  Distance,  Motion,  Force, 
and  Convergence/ ;  the  qualities  or  principles  of  which  I  will 
define  as  follows :  Time  always  has  direct  reference  to  the  Period, 
namely  the  periodic  time,  or  entire  revolution  of  the  planet 
from  any  given  point  in  its  orbit,  to  the  same  point  again ;  which 
period  or  periodic  time,  may  be  divided  and  subdivided,  as  occa- 
sion may  require,  when  comparing  other  elements  with  that  of 
time  ;  and  hence  the  period  or  whole  periodic  time  of  a  planet, 
as  also  the  mean  distance  of  a  planet  from  the  centre  of  grav- 
ity may,  in  contradistinction  to  the  elements  of  Motion,  Force, 
and  Convergency,  properly  be  called  positive  quantities,  by  which, 
what  may  be  called  the  rates  (ratios  or  proportions)  of  the  ele- 
ments of  motion,  force  and  convergency  may  be  compared. 

Distance,  or  the  mean  distance  of  a  planet,  may  always  be 
abstractly  expressed  by  some  numerical  quantity,  by  which 
the  mean  distance  of  any  given  planet  of  a  system  is  propor- 
tioned to  the  mean  distance  of  that  planet  of  the  system  which 


ON    PLANETARY    FORCE    AND    MOTION.  199 

is  assumed  at  unity;  thus,  if  the  mean  distance  of  our  earth 
from  the  sun  be  assumed  at  unity  or  1,  the  mean  distance  of 
Mercury  will  be  expressed  by  the  fraction  .38710 ;  that  of  Ve- 
nus, by  the  fraction  .72332 ;  that  of  Mars,  by  unity  plus  the 
fraction  .52998,  &c.  So  also  if  the  orbit  be  eccentric,  we  speak 
of  the  mean  distance,  the  perihelion  distance,  &c. 

The  mean  distance  of  a  planet  determines  or  denotes  the  cir- 
cumference' of  the  orbit,  and  consequently,  the  amount  of  space 
passed  over  in  an  entire  period.  It  also  determines  the  dilata- 
tion of  the  orbit,  which  is  consequently  as  the  mean  distance. 
But  the  dilatation  of  the  orbit  which  is  denoted  by  the  mean 
distance,  must  not  be  confounded  with  the  rate  of  convergency 
of  the  planet,  which  is  denoted  by  the  time  of  the  period.  Mo- 
tion, which  I  shall  use  as  synonymous  with  Velocity,  de- 
notes a  passing  over  space ;  and  as  this  takes  place  in  time,  it 
also  denotes  the  velocity  or  rapidity  with  which  space  is  passed 
over. 

Motion  manifestly  depends  upon  the  force  of  gravity,  at 
least,  in  retaining  the  planet  in  its  orbit ;  and  if  the  force  of 
gravity  be  inversely  as  the  distance,  (as  I  contend,)  the  rate  of 
mean  motion  will  always  be  inversely  as  the  whole  amount  of  the 
force  of  gravity  expended  during  the  entire  period  of  the  planet. 

Force,  which  I  shall  use  as  synonymous  with  gravity,  or  the 
force  of  gravity,  denotes  the  power  of  attraction  by  which  one 
body  is  drawn  towards  another,  causing  the  tendency  of  one 
body  towards  another ;  as  that,  of  a  planet  towards  the  sun,  or  a 
satellite  towards  its  primary ;  the  intensity,  or  rather,  proportional 
intensity  of  which  manifestly  depends  upon  the  proportional  dis- 
tance ;  and  it  increases  or  decreases  in  the  inverse  ratios  of 
some  given  root  or  power  of  the  distance ;  which  intensity  I 
shall  in  general  denominate  the  rate  of  gravity. 

And  as  by  my  determination  of  the  law  of  gravity,  the  rate  of 
mean  motion  will  always  be  found  to  be  inversely  as  the  whole 
amount  of  the  force  of  gravity  expended  during  the  entire  period 
of  the  planet,  so  also,  by  the  same  law,  the  rate  of  mean  force 
will  always  be  inversely  as  the  whole  amount  of  motion  during 
an  entire  period  of  the  planet, — thus  reciprocating  with  each 
other  upon  perfectly  equal  and  mutual  terms. 

Convergency  is  measured  by  degrees,  minutes,  &c.  of  the 
circle,  and  hence  the  amount  of  convergency  in  the  time  of  the 
period  of  any  planet  in  360° ;  consequently  the  amount  of  con- 
vergency in  any  one  orbit  is  properly  unity. 

Convergency  is  no  part  of  the  motion  of  a  planet ;  but  simply 
the  direction  of  a  motion;  nevertheless,  there  could  be  no  con- 
vergency without  motion  ;  and  although  convergency  is  produc- 


200  ON    PLANETARY    FORCE    AND    MOTION. 

ed  by  constant  gravity,  we  perceive  that  motion  is  as  essential 
to  this  element  as  gravity  is.  Hence,  motion  and  convergency 
seem  to  be  alike  intimately  connected  with,  and  dependent  upon 
gravity,  for  the  exhibition  of  their  phenomena.  Nor  is  it  certain 
that  motion  is  an  element  that  is  any  more  independent  of  grav- 
ity, than  convergency  is,  or  that  gravity  is  any  less  attentive  to 
motion  than  it  is  to  convergency,  —  notwithstanding  those  notions 
drawn  from  the  Newtonian  theory  of  the  universal  and  equal 
gravity  of  matter. 

But  this  theory  of  universal  gravity  being  established,  and 
inasmuch  as  gravity  was  to  have  little  or  nothing  to  do  with 
motion,  otherwise  than  to  accelerate  or  retard  it,  it  was  neces- 
sary that  a  planet  should  be  projected  in  a  tangent  to  some  point 
of  its  orbit,  thereby  giving  motion  ;  which,  whether  it  were  more 
or  less,  would  be  taken  charge  of  by  the  innate  gravity  of  distant 
matter.  And  as  such  force  would  not  be  likely  to  produce  a 
motion  just  sufficient  to  balance  the  force  of  gravity,  those  ele- 
ments were  left  to  keep  the  balance  of  power,  by  alternately 
getting  the  better  of  each  other ;  which  contrivance,  however, 
was  found  insufficient,  until  a  centrifugal  force  was  finally 
attached  to  the  machinery,  endued  with  such  forces  as  might  be 
found  necessary  to  accomplish  the  object. 

Next,  the  planets  were  found  to  be  of  different  densities,  in 
proportion  to  their  respective  bulks  or  magnitudes ;  because  it 
was  found  they  did  not  possess  the  same  power  of  attraction  in 
proportion  to  their  magnitudes ;  and  notwithstanding,  had  anal- 
ogy been  applied  to  the  train  of  facts  which  bear  upon  that  ques- 
tion, such  hypothesis  would  long  since  have  been  overthrown. 
But  having  adopted  the  Newtonian  theory  of  innate  and  univer- 
sal gravity,  as  the  first  great  article  of  faith,  neither  fact  nor 
reason  has  made  inroads  upon  it. 

The  hypothesis  of  universal  gravity  being  thus  far  established 
and  put  at  rest  beyond  a  wish  or  disposition  to  doubt,  to  ascer- 
tain the  ratio  which  the  force  of  gravity  or  attraction  bears  to  the 
distance  from  the  attracting  body,  became  a  disideratum ;  and 
M.  Vince  says,  that  Sir  Isaac  Newton  gave  Dr.  Halley  "  a  proof" 
of  the  law  of  gravity  as  proportioned  to  distance,  namely,  that  it 
varies  inversely  as  the  square  of  the  distance.  And  this  hypo- 
thesis is  supposed  to  have  been  clearly  deduced  from  a  compar- 
ison of  the  law  of  falling  bodies  near  the  earth,  with  the  rate  of 
deflection  of  the  moon  from  a  tangent  to  its  orbit. 

Thus,  in  lieu  of  resorting  to  the  natural  elements  of  the 
planets  for  the  purpose  of  determining  this  very  important  ques- 
tion, they  have  created  for  the  purpose,  the  factitious  element  of 
deflection,  so  wholly  extraneous  and  foreign  to  the  laws  of  force 


ON    PLANETARY    FORCE    AND    MOTION.  201 

and  motion,  as  to  have  been  a  very  troublesome  element  to 
astronomers  and  mathematicians  ;  for  having  suffered  their  imag- 
inations to  run  off  in  a  tangent  to  the  orbit,  where  no  force  nor 
motion  is,  (though  but  a  very  little  distance,)  in  accordance  with 
the  precaution  from  the  highest  authority,  yet  if  a  mathematical 
error  be  contained  in  such  very  little  distance,  it  may,  nevertheless, 
in  its  consequences,  be  exceeding  broad. 

Any  amount  of  deflection  is  conceived  to  be  measured  or 
denoted  by  direct  linear  quantity,  namely,  by  a  perpendicular 
line  from  the  given  tangent  to  the  revolving  body  in  the  orbit ; 
and  hence,  if  the  rate  of  progression  of  the  supposed  fall  or 
deflection  of  the  planet  from  the  tangent,  be  ascertained  or  estab- 
lished, it  could  not  well  be  made  to  apply  to  the  entire  period  or 
revolution  of  the  planet;  nor  could  the  whole  amount  of  deflec- 
tion in  the  time  of  the  period  be  well  ascertained,  which  should 
be  the  case  with  any  element  which  may  properly  be  said  to 
have  its  rate,  namely,  its  ratio  to  that  of  other  elements  of  the 
same  planet. 

Nevertheless,  what  may  be  termed  the  element  of  deflection, 
is  by  no  means  as  useless  as  that  of  a  centrifugal  force,  but  is 
susceptible,  when  rightly  considered,  of  furnishing  or  throwing 
much  light  upon  the  law  of  gravity  in  respect  to  distance  ;  and 
this  too  by  comparing  it  with  the  law  of  falling  bodies  near  the 
earth ;  in  which  comparison,  we  find  many  things  in  which 
they  corroborate  each  other;  and  hence,  the  identity  of  the 
force  which  causes  a  body  to  descend  or  fall  to  the  earth,  and 
that  which  retains  the  moon  in  its  orbit,  goes  in  no  sense  to 
prove,  that  the  deflection  of  a  planet  from  a  tangent,  proceeds 
by  the  same  progression  as  that  of  a  falling  body  from  a  state  of 
rest,  namely,  by  a  progression  which  shall  be  denoted  by  the  suc- 
cessive odd  numbers,  as  1,  3,  5,  &c. 

For  if  so,  then  the  amount  of  deflection  of  a  planet  whose 
distance  is  4,  when  it  has  revolved  over  l-8th  of  its  orbit,  should  be 
equal  to  the  amount  of  deflection  of  a  planet  of  the  same  system 
whose  distance  is  1,  when  it  has  revolved  over  l-4th  of  its 
orbit ;  for  the  motion  of  the  planet  whose  distance  is  1,  must 
be  twice  as  great  as  that  whose  distance  is  4,  for  otherwise  the 
squares  of  their  periods  could  not  be  as  the  cubes  of  their  mean 
distances  from  the  sun. 

But  in  lieu  of  the  rates  of  deflection  being  equal,  when  that 
whose  distance  is  1,  has  passed  over  l-4th  of  its  orbit,  and  that 
whose  distance  is  4,  has  passed  over  l-8th  of  its  orbit,  their  amounts 
of  deflection  will  then  be  proportioned  to  each  other,  as  1  1o 
1.17156,  and  hence  the  amount  of  deflection  of  a  planet  while 
revolving  over  l-4th  of  its  orbit  will  be  2.41421  times  as  great  in 
26 


202  ON    PLANETARY    FORCE    AND    MOTION. 

the  last  half  of  the  time,  as  in  the  first  half;  when  by  the  law  of 
falling  bodies,  it  should  be  just  three  times  as  great  in  the  last 
half  of  the  time  as  in  the  first  half. 

I  am  aware  that  Sir  Isaac  Newtonand  others  have  said,  that 
if  from  the  point  of  intersection,  or  touch  of  the  given  tangent 
with  the  orbit,  we  take  a  very  little  distance,  the  deflection  will  be 
likened  to,  or  rather  be  identical  in  its  mathematical  construction 
with  the  law  of  falling  bodies  near  the  earth  ;  which  allegation 
has  certainly  supplied  the  place  of  mathematical  certainty  for  a 
long  time.  But  surely  we  ought  to  have  been  informed  where 
the  little  distance  should  stop ;  for  as  the  progress  of  deflection 
in  any  assignable  portion  of  its  orbit,  is  different  from  that  of  a 
falling  body  near  the  earth,  it  would  seem  necessary,  in  order 
that  such  little  distance  might  present  a  mathematical  truth,  that 
it  should  be  less  than  any  assignable  quantity,  namely,  in  its 
ultimate  ratio,  alias  in  0,  or  no  distance  whatever ;  in  which 
point  or  place,  far  too  many  important  questions  have  been  set- 
tled ;  as  if  ratios  could  as  well  exist  in  0,  as  in  actual  numerical 
quantities  ;  which,  in  point  of  demonstration,  would  seem 
too  much  to  resemble  those  important  discoveries  we  have 
somewhere  read  of,  by  which  many  of  the  laws  of  gravity  and 
attraction  were  deduced  and  explained,  by  geometrizing  upon 
two  inconceivably  small  corpuscles,  or  physical  atoms. 

I  will  now  return  to  a  further  consideration  of  convergency, 
and  the  other  proper  elements  of  the  planets. 

Convergency  is  most  directly  connected  with  the  phenomena 
of  motion  in  the  heavenly  bodies,  and  with  the  law  of  gravity ; 
and  being  the  description  or  delineation  of  the  degrees,  minutes, 
seconds,  &c.  of  a  circle,  produced  in  time,  which  becomes  full 
and  complete  in  the  time  of  the  period,  hence,  the  rate  of  conver- 
gency is  inversely  as  the  time  of  the  period ;  and  the  amount  of 
convergency  in  the  time  of  the  period,  is  in  all  cases  the  same, 
namely,  360°. 

Thus,  while  time,  or  rather  the  Period,  and  Distance,  may  be 
called  the  positive  elements  of  the  planets,  those  of  Motion, 
Gravity,  and  Convergency,  may  properly  be  called  the  relative 
elements ;  performing  relative  operations  in  respect  to  each 
other,  and  also  in  respect  to  Period  and  Distance. 

The  relative  elements  are  therefore  properly  said  to  have  their 
rates  of  operation  ,  not  only  in  respect  to  each  other,  but  in  re- 
spect to  time  or  period,  and  distance ;  which  rates  of  operation 
become  complete  and  entire  in  the  time  of  the  period,  which 
makes  what  may  be  called  the  whole  amount  of  the  operation. 
And  thus,  Motion,  Gravity  and  Convergency,  are  said  to  have 
both  their  rates  and  amounts  of  operation. 

The  period  of  a  planet  is  the  cube  of  the  square  root  of  the 


ON    PLANETARY    FORCE    AND    MOTION.  203 

mean  distance ;  so  then  the  mean  distance  is  the  square  of  the 
cube  root  of  the  period ;  for  if  it  were  otherwise,  the  square  of 
the  period  of  a  planet  could  not  be  the  cube  of  the  mean  dis- 
tance ;  nor  could  the  squares  of  the  periods  of  a  system  of 
planets  be  as  the  cubes  of  their  mean  distances  from  the  sun. 
And  that  the  rates*  and  amounts  of  motion,  gravity,  and  conver- 
gency,  are  so  proportioned  to  each  other,  and  to  the  period  and 
distance,  as  to  produce  that  grand  proportional  result,  is  what 
no  one  will  question  or  doubt. 

Hence  the  rate  of  mean  motion  of  a  planet  is  inversely  as  the 
cube  root  of  the  period,  or  the  square  root  of  the  mean  distance; 
and  the  amount  of  motion  in  the  time  of  the  period,  is  as  the 
mean  distance  ;  for  the  mean  distance  denotes  or  represents  the 
circumference  of  the  orbit,  and  consequently  the  amount  of 
space  passed  over;  which  properly  denotes  or  determines  the 
amount  of  motion.  Hence,  the  mean  rate  of  motion  is  to  the 
whole  amount  of  motion,  as  unity  is  to  the  time  of  the  period. 

The  rate  of  gravity  or  force,  is  inversely  as  some  power  or 
root  of  the  distance  ;  which  I  assume  to  be  inversely  as  the  first 
power,  or  first  root,  of  the  distance,  viz.,  inversely  aa  the  dis- 
tance. Hence,  the  mean  rate  of  gravity  of  a  planet  is  inversely 
as  the  mean  distance,  and  also  inversely  as  the  square  of  the 
cube  root  of  the  period,  and  is  consequently  the  square  of  the 
mean  rate  of  motion.  The  whole  amount  of  gravity,  then,  in  the 
time  of  the  period,  will  be  the  square  root  of  the  distance,  which 
is  also  the  cube  root  of  the  period ;  and  hence  the  mean  rate 
of  gravity  is  to  the  whole  amount  of  gravity,  as  unity  is  to  the 
time  of  the  period. 

The  mean  rate  of  convergency  is  inversely  as  the  time  of  the 
period,  or  inversely  as  the  cube  of  the  square  root  of  the  dis- 
tance ;  and  hence,  is  the  cube  of  the  mean  rate  of  motion,  and 
the  cube  of  the  square  root  of  the  mean  rate  of  gravity  or  force. 
Consequently,  the  whole  amount  of  convergency  in  the  time  of 
the  period  is  unity  or  1 ;  and  hence,  the  mean  rate  of  conver- 
gency is  to  the  whole  amount,  as  unity  is  to  the  time  of  the 
period ;  or  rather,  as  the  inverse  of  the  time  of  the  period  is  to 
unity. 

Hence,  the  mean  rate  of  motion,  of  gravity,  or  of  conver- 
gency, is  to  the  whole  amount  in  the  time  of  the  period,  as  unity 
is  to  the  time  of  the  period. 

Hence,  the  product  of  the  mean  rate  of  motion  by  the  time 
of  the  period,  gives  the  whole  amount  of  motion  in  the  time  of 
the  period. 

The  product  of  the  mean  rate  of  gravity  or  force,  by  the  time 
of  the  period,  gives  the  whole  amount  of  gravity  or  force  in  the 
time  of  the  period. 


204  ON    PLANETARY    FORCE    AND    MOTION. 

And  the  product  of  the  mean  rate  of  convergency  by  the  time 
of  the  period,  gives  the  whole  amount  of  convergency  in  the  time 
of  the  period. 

Thus  the  mean  rate  of  gravity  of  a  planet  is  always  a  mean 
proportional  between  the  mean  rate  of  motion,  and  the  mean 
rate  of  convergency ;  and  is  equally  attentive  to  both.  Nor  does 
it  ever  forsake  its  position,  or  neglect  its  trust ;  for  whether  the 
orbit  be  centric  or  eccentric,  the  element  of  gravity  at  all  times, 
and  in  all  parts  of  the  orbit,  preserves  the  same  position  in  re- 
spect to  the  elements  of  motion  and  convergency,  viz.,  a  mean 
proportional  between  them,  and  hence,  the  planets  sing  through 
the  heavens  with  as  much  harmony  and  unity  of  purpose,  as 
we  can  desire. 

Kepler,  in  addition  to  the  great  law,  "  That  the  squares  of  the 
periods  of  the  planets,  are  as  the  cubes  of  their  mean  distances 
from  the  sun,"  also  discovered  the  important  fact,  "  That  the 
motion  of  an  individual  planet  revolving  in  an  eccentric  orbit, 
varies  in  the  inverse  ratio  to  that  of  its  varying  distance  from 
the  sun."  But  Kepler  did  not  discover  that  the  rates  of  motion 
of  the  planets  are  inversely  as  their  distances  from  the  sun  ; 
although  such  has  often  been  the  version  and  the  understand- 
ing in  respect  to  his  discovery.  And  it  requires  but  little  con- 
sideration to  convince  any  one,  that  the  rates  of  motion  of  the 
planets  are  not  inversely  as  their  mean  distances  from  the  sun  ; 
for  if  it  were  so,  the  squares  of  the  periods  of  the  planets  would 
be  as  the  fourth  powers  of  their  mean  distances  from  the  sun ;  or 
in  other  words,  the  time  of  their  periods  would  be  as  the  squares 
of  their  mean  distances  from  the  sun  ;  as  any  one  will  discover 
on  a  moment's  reflection.  For  if  the  mean  distance  were  4,  and 
the  mean  motion  were  inversely  as  the  mean  distance,  the  period 
would  be  16,  in  lieu  of  8.  And  hence,  such  hypothesis  is  un- 
true. 

And  yet,  this  erroneous  version  of  Kepler's  law,  has  been  a 
great  supporter  of  the  hypothesis,  that  the  rate  of  gravity  is  in- 
versely as  the  square  of  the  distance ;  which  error  was  com- 
mitted by  Sir  Isaac  Newton,  in  referring  the  square  of  the  dis- 
tance to  the  rate  or  intensity  of  gravity,  in  lieu  of  referring  it  to 
the  rate  of  deflection,  or  to  the  result  or  effect  of  the  gravity. 

Thus,  once  having  adopted  the  well  established  fact  in  dy- 
namics, that  twice  the  motion  requires  quadruple  the  force  to 
retain  a  planet  in  its  orbit,  it  necessarily  follows  as  a  corollary  or 
consequent,  that  if  the  mean  motions  of  the  planets  of  a  system, 
are  inversely  as  their  mean  distances  from  the  sun,  the  force  of 
gravity  which  retains  them  in  their  orbits,  must,  of  necessity,  be 
inversely  as  the  square  of  the  distance  from  the  sun.  And  the 


ON    PLANETARY    FORCE    AND    MOTION.  205 

next  corollary  or  consequent,  which  as  necessarily  follows,  is, 
that  their  periods  will  be  as  the  squares  of  their  mean  distances 
from  the  sun ;  or  that  the  squares  of  their  periods  will  be  as  the 
fourth  powers  of  their  mean  distances  from  the  sun  ;  and  also  that 
the  period  of  each  planet  of  the  system  will  be  the  square 
of  its  mean  distance  from  the  sun,  by  which,  error  begets  error. 

And  the  same  or  like  consequences  would  follow,  in  a  sys- 
tem of  eccentric  orbits,  that  would  in  a  system  of  centric  orbits, 
viz.,  if  the  mean  motions  of  the  planets  were  inversely  as  their 
mean  distances  from  the  sun;  and  also  if  from  the  points  of 
mean  motion,  the  motion  of  the  planet  does  vary  in  the  inverse 
ratio  as  the  distance  from  the  sun  varies,  as  Kepler  has  shown ; 
then  their  motions  throughout  their  respective  orbits  would  be 
inversely  as  their  distances  from  the  sun ;  presenting  precisely 
the  same  or  similar  results  in  this  respect,  as  would  be  present- 
ed in  a  system  of  centric  orbits,. viz.,  the  period  of  each  would 
be  the  square  of  the  mean  distance  ;  and  the  periods  of  the  re- 
spective planets  of  the  system,  would  be  as  the"  squares  of  their 
mean  distances  from  the  sun  ;  or  the  squares  of  their  periods 
would  be  as  the  fourth  powers  of  their  mean  distances  from  the 
sun.  And  if  such  were  the  case,  the  consequence  would  con- 
clusively follow,  that  the  force  of  gravity  would  be  inversely  as 
the  square  of  the  distance  from  the  sun. 

But  Kepler  taught  no  such  doctrine ;  he  did  not  attempt  so 
readily  to  abrogate  and  make  void  his  great  law,  so  well  founded 
in  observation  and  in  the  powers  and  roots  of  numbers,  viz.,  that 
the  squares  of  the  periods  of  a  system  of  planets  are  as  the 
cubes  of  their  mean  distances  from  the  sun ;  nor  to  overthrow 
the  well  established  fact,  that  twice  the  motion  requires  quadru- 
ple the  force,  when  he  thus  declared  the  no  less  important  dis- 
covery :  That  the  motion  of  a  planet,  viz.,  an  individual  planet, 
revolving  in  an  eccentric  orbit,  varies  from  the  points  of  mean 
motion,  in  the  inverse  ratio  as  the  distance  varies  from  the  sun. 

I  say  that  the  discovery  by  Kepler,  —  that  in  an  eccentric  orbit, 
the  motion  varies  from  the  points  of  mean  motion  in  the  inverse 
ratio  as  the  distance  from  the  sun  varies,  —  is  no  less  important 
than  are  his  other  discoveries.  For  if  it  were  otherwise,  the 
revolution  of  a  planet  in  an  eccentric  orbit  would  be  wholly  in- 
explicable. . 

It  is  this  principle  that  chains  the  planet  in  its  eccentric  orbit, 
and  keeps  it  obedient  to  the  power  of  gravity,  as  much  so  in 
one  part  of  the  orbit  as  in  another ;  and  hence  prevents,  or  avoids 
the  fortuitous  necessity,  that  gravity  should  get  the  better  of  the 
motion  at  aphelion;  and  that  the  motion  should  get  the  better  of 
gravity  at  perihelion,  in  order  to  preserve  the  balance  of  power, 


206  ON    PLANETARY    FORCE    AND    MOTION. 

by  the  help  of  a  centrifugal  force ;  as  the  laws  of  gravity  and 
motion  have  been  taught  and  explained  by  Maclaurin,  Fergu- 
son and  others,  to  a  world  of  too  easy  faith. 

Hence,  it  will  be  found,  (as  I  propose  to  show  in  the  sequel,) 
that  both  the  gravity  and  convergency  of  a  planet  revolving  in 
an  eccentric  orbit,  vary  from  their  mean,  in  the  same  ratio  that 
motion  does,  viz.,  in  the  inverse  ratio  as  the  distance  varies  from 
the  mean,  or  from  the  sun ;  which  is  one  and  the  same  thing. 
And  hence,  the  rate  of  gravity  always  remains  a  mean  pro- 
portional between  the  rate  of  motion  and  the  rate  of  con- 
vergency, controlling  both  of  them,  as  well  in  one  part  of  the 
orbit  as  in  another. 

But  in  -respect  to  the  Newtonian  law  of  gravity,  viz.,  that 
gravity  is  inversely  as  the  square  of  the  distance,  nothing  can  be 
more  manifest,  than  that,  in  order  to  sustain  such  law,  the  mean 
motion  of  any  planet  of  a  system,  whether  revolving  in  a  centric 
or  eccentric  orbit,  must,  in  reference  to  the  mean  motions  of  the 
other  planets  of  the  system,  be  inversely  as  the  mean  distance, 
which  is  not  the  case;  but  in  such  case,  the  squares  of  the 
periods  of  the  planets  would  be  as  the  fourth  powers  of  their 
mean  distances  from  the  sun. 

SECTION    SECOND. 

That  the  apparent  motions  of  the  planets  are  not  equable  in 
their  orbits,  seems  to  have  been  familiarly  known  to  ancient 
astronomers ;  and  hence,  that  the  orbits  in  respect  to  the  earth, 
as  was  then  supposed,  were  eccentric. 

This  seems  to  have  been  the  opinion  of  Ptolemy  in  ancient 
times,  and  of  Tycho  in  more  modern  times ;  and  upon  this  hy- 
pothesis they  endeavored  to  equate  their  motions;  but  when 
it  became  more  rationally  understood,  that  the  eccentricity  of  the 
orbits  referred  to  the  sun  directly,  and  not  to  the  earth,  the  idea 
that  each  planet  revolved  around  some  point  in  its  orbit,  about 
which  the  motion  was  equable,  was  of  consequence  abandoned, 
and  gave  place  to  new  researches  into  the  phenomena  of  the 
motions  of  planets,  preparatory  to  accounting  for  such  eccen- 
tricity. 

In  this  respect,  Kepler,  who  always  seemed  preeminent  in 
searching  out  physical  cauess  and  phenomena,  discovered  the 
fact,  —  which  is  perhaps  no  less  important  in  respect  to  eccentric 
orbits,  than  is  his  great  law  that  the  squares  of  the  periods  of  a 
system  of  planets,  are  as  the  cubes  of  their  mean  distances  from 
the  sun,  in  respect  to  orbits  in  general,  — -  that  the  motion  of 
a  planet  revolving  in  an  eccentric  orbit,  varies  inversely  as  the 


ON  PLANETARY  FORCE  AND  MOTION.  207 

distance  from  the  sun  varies ;  which  law  is  so  essential  and 
important  in  respect  to  the  motion  of  a  planet  in  an  eccentric 
orbit,  that  any  attempt  to  account,  without  the  law,  or  to  attempt 
to  supply  its  place  by  extraneous  matter,  would  to  me  appear 
like  a  forlorn  hope.  And  why  this  law,  in  connection  with  the 
other  great  law  just  quoted,  should  not  long  ere  this  have  dis- 
closed more  important  truths  in  respect  to  the  laws  of  force  and 
motion  than  they  have  done,  appears  to  me  wholly  unaccounta- 
ble. 

It  will  be  readily  seen  by  any  one  who  will  examine  the  sub- 
ject but  a  little,  that  a  misapprehension  of  said  law,  viz.,  that  the 
motion  of  a  planet  varies  in  its  orbit  inversely  as  the  distance 
from  the  sun  varies,  would  lead  to,  and  produce  the  same  in- 
explicable difficulties  which  now  pervade  the  science  of  astron- 
omy ;  and  that  said  law  has  been  wholly  misapprehended,  there 
perhaps  is  but  little  doubt. 

If  the  mean  motions  of  a  system  of  planets  (whether  com- 
posed of  centric  orbits,  eccentric  circles,  or  ellipses)  were  in- 
versely as  their  mean  distances  from  the  sun,  and  if  also  in  each 
eccentric  orbit,  the  motion  should  vary  from  the  point  of  mean 
motion  inversely  as  the  distance  from  the  sun  should  vary  while 
revolving  in  its  orbit,  in  such  case  nothing  can  be  made  more 
manifest ;  no  more  simple  proposition  exists  than  that  force  or 
gravity  must  of  necessity  be  inversely  as  the  square  of  the  dis- 
tance. And  it  is,  also,  every  way  as  simple  a  manifestation, 
that  if  such  were  the  case,  Kepler's  other  great  law  would  be 
made  wholly  void ;  for  in  such  case,  the  squares  of  the  periods 
of  a  system  of  planets,  would  be  as  the  fourth  powers  of  their 
mean  distance  from  the  sun. 

In  such  case,  the  mean  motion  of  a  planet  whose  distance  is 
4,  would  have  to  be  .25,  —  when  we  know  it  must  be  .5  in  order 
to  fulfil  the  great  law.  And  the  only  possible  condition,  as  I 
have  already  shown,  upon  which  the  hypothesis  that  the  force 
is  inversely  as  the  square  of  the  distance,  could  be  even  hypo- 
thetically  sustained,  would  be  upon  the  ground,  that  the  mean 
motions  of  a  system  of  planets  are  inversely  as  their  mean 
distances  from  the  sun ;  and  also,  that  the  motion  of  any  planet 
revolving  in  an  eccentric  orbit,  varies  from  the  point  of  mean 
motion,  inversely  as  the  distance  from  the  sun  varies. 
,  But  Kepler  adopted  but  the  last  part  of  said  hypothesis,  viz., 
that  a  planet  revolving  in  an  eccentric  orbit  does  vary  from  the 
points  of  mean  motion  inversely  as  the  distance  of  the  revolving 
body  varies  from  the  sun ;  and  this  permits  his  other  law  to 
stand,  and  all  the  elements  of  the  planets  to  operate  in  unison. 

The  mean  motions  of  the  planets  then,  are  inversely  as  the 


208  ON    PLANETARY    FORCE    AND    MOTION. 

square  roots  of  their  mean  distances  from  the  sun;  that  the 
cubes  of  their  mean  distances  may  be  as  the  squares  of  their 
periods. 

But  what  has,  perhaps,  served  much  to  sustain  mankind  in 
their  misapprehension  of  Kepler's  law,  is  that  useless  demon- 
stration, (so  far  as  astronomy  is  directly  concerned  or  ever  has 
been,)  that  if  a  revolving  body  vary  in  its  motion  inversely 
as  its  distance  varies  from  the  sun,  it  will  describe  or  pass  over 
equal  areas  in  equal  times.  This  demonstration,  although  true, 
has  never  been  used  in  astronomy  to  any  other  effect  than  "to 
perplex  and  dash  maturest  counsels." 

In  -any  system  of  planets,  whether  all  the  orbits  of  such  sys- 
tem are  centric,  eccentric,  or  elliptical,  if  their  mean  motions 
were  inversely  as  their  mean  distances  from  the  sun,  the  planets 
of  such  system  would  in  respect  to  each  other  describe  equal 
areas  in  equal  times. 

In  such  case,  the  mean  motion  of  the  planet  whose  mean 
distance  js  4,  would  be  .25,  or  half  what  we  know  to  be  its  ac- 
tual motion ;  and  its  period  would  be  16,  or  the  square  of  its 
distance,  or  twice  its'  actual  period ;  and  the  area  passed  over 
in  the  time  of  its  period,  would  be  16  times  as  great  as  that 
passed  over  by  the  planet  whose  distance  is  1,  in  the  time  of  its 
period.  That  is,  in  such  case,  the  areas  passed  over  by  the 
several  planets  of  a  system,  would  be  inversely  as  their  motions ; 
or  would  be  as  the  square  roots  of  their  respective  distances ; 
and  hence,  would  be  equal  in  equal  times.  Then,  indeed,  would 
the  force  be  inversely  as  the  square  of  the  distance ;  and  the 
motion  would  be  the  fourth  root  of  the  force ;  the  squares  of 
the  periods  of  a  system  of  planets,  would  be  as  the  fourth  pow- 
ers of  their  mean  distances  from  the  sun  ;  or  their  periods  would 
be  as  the  squares  of  their  mean  distances  from  the  sun. 

If  the  distance  of  the  earth  from  the  sun  were  assumed  at  1, 
the  distance  of  a  planet  whose  period  was  8,  would  be  the 
square  root  of  8,  in  lieu  of  the  square  root  of  16,  as  now  sup- 
posed. Thus,  the  distances  of  the  planets  of  our  system  would 
be  reduced  from  that  of  the  third  roots  of  the  squares  of  their 
respective  periods,  to  that  of  the  square  roots  of  their  respective 
periods.  The  distance  of  Herschel  from  the  sun  would  be 
much  less  than  half  the  distance  it  has  heretofore  been  estimated 
at,  even  though  the  earth  were  distant  95,000,000  miles.  But  as 
Venus  would  have  a  greater  proportional  distance  from  the  sun, 
in  proportion  to  that  of  the  earth,  than  has  been  assumed,  (un- 
less from  its  greatest  'elongation  or  otherwise,  its  true  propor- 
tional distance  has  been  determined,)  hence,  the  actual  distance 
of  the  earth  from  the  sun  in  miles,  would  have  been  overrated, 


ON    PLANETARY    FORCE    AND    MOTION.  209 

in  which  case  the  actual  distance  of  all  the  superior  planets 
would  have  been  overrated  in  like  proportion. 

So  it  seems,  in  such  case,  the  superior  planets  would,  in  a 
measure,  be  relieved  from  the  charge  of  a  lack  of  density  in 
proportion  to  their  respective  bulks  or  magnitudes,  which  has 
be§n  charged  against  them  by  some ;  as  their  bulks  or  magni- 
tudes would  be  vastly  less  than  had  been  supposed ;  with  an 
innumerable  train  of  consequences  which  I  will  not  here  at- 
tempt to  follow  out. 

It  then  becomes  conclusive,  that  if  the  mean  motions  of  the 
several  planets  of  a  system  are  inversely  as  the  mean  distances 
from  the  sun,  the  periods  of  the  planets  will  be  as  the  squares 
of  their  mean  distances  from  the  sun. 

But  Kepler  only  intended  to  declare,  and  only  did  declare, 
that  the  motion  of  any  individual  planet  of  a  system  varies 
from  the  point  of  mean  motion  (if  the  orbit  be  eccentric)  in- 
versely as  the  distance  varies  from  the  sun:  In  which  case 
the  radius  vector  of  such  planet  will  pass  over  or  describe  equal 
areas  in  equal  times ;  notwithstanding  the  mean  motions  of  all 
the  planets  of  the  system  are  inversely  as  the  square  roots  of 
their  respective  distances.  And  in  which  case  also,  the  great 
law  that  the  squares  of  their  periods  are  as  the  cubes  of  their 
mean  distances,  will  still  apply  to  the  system  in  all  its  force  and 
majesty. 

I  shall  adopt,  fully,  Kepler's  two  said  laws  as  he  gave  them ; 
and  consequently  adopt  the  hypothesis,  that  the  force  or  gravity, 
of  any  individual  planet,  varies  inversely  as  the  distance  varies 
from  the  sun ;  as  that  of  necessity  follows  upon  the  adoption  of 
said  two  laws.  And  it  is  only  upon  the  adoption  of  those  prin- 
ciples, that  we  are  enabled  in  anywise  to  account  for  the  mo- 
tions of  a  system  of  planets  revolving  in  eccentric  orbits.  And 
that  the  primary  planets  of  the  solar  system  do  all  revolve  in 
eccentric  orbits,  is  doubtless  true ;  and  that  such  is  the  case  with 
all  the  heavenly  bodies,  is,  at  least,  more  than  probable. 

The  adoption  then  of  Kepler's  law,  that  the  squares  of  the 
periods  of  a  system  of  planets  are  as  the  cubes  of  their  mean 
distances  from  the  sun ;  and  also,  his  law  which  teaches  that  the 
motion  of  a  planet  revolving  in  an  ecceniric  orbit  varies  from 
the  points  of  mean  motion  inversely  as  the  distance  varies  from 
the  sun  in  the  course  of  its  revolution,  together  with  the  neces- 
sary and  imperious  consequent  upon  said  law,  viz.  that  the 
force  of  gravity  varies  inversely  as  the  distance  varies  from  the 
sun,  —  furnishes  a  very  easy  and  simple  solution,  accounting 
for  the  movements  of  a  system  of  planets  in  eccentric  orbits ; 
and  we  shall  hence  discover  that  those  principles  are  entirely 


210  ON    PLANETARY    FORCE    AND    MOTION. 

able  to  manage  a  planet  throughout  its  entire  period,  without 
the  aid  of  any  other  machinery  whatever. 

I  will  suppose  a  system  of  planets  revolving  around  the  sun, 
or  around  one  and  the  same  central  power ;  and  that  as  occa- 
sion may  require,  for  the  purposes  of  explanation,  some  of  the 
planets  may  be  conceived  to  revolve  in  centric  orbits,  and  others 
in  eccentric  orbits,  being  nevertheless  circles. 

The  line  of  the  apsides  is  a  right  line  extending  through  the 
centre  of  the  sun,  from  the  perihelion  to  the  aphelion ;  hence  the 
time  of  the  apsides  will  be  the  diameter  of  the  orbit,  and  may 
be  thus  denoted,  even  in  a  centric  orbit ;  although  such  orbit 
cannot  properly  be  said  to  have  perihelion  and  aphelion. 

In  such  system  of  planets,  the  squares  of  their  periods  would 
be  as  the  cubes  of  their  mean  diameters  from  the  sun.  For  as 
it  is  not  necessary  in  order  for  a  system  of  planets  to  fulfil  this 
law,  that  the  orbits  should  be  alike  eccentric,  hence  if  some  of 
them  were  centric  and  others  eccentric,  they  would  equally  well 
fulfil  the  law;  hence,  if  the  lines  of  apsides  of  an  eccentric 
orbit  and  of  a  centric  orbit  of  the  system  were  equal,  the  two 
planets  would  pass  over  equal  areas  in  the  times  of  their  re- 
spective periods;  the  motion  of  one  being  uniform,  and  that 
of  the  other  constantly  varying  inversely  as  the  distance  from 
the  sun  varies. 

It  will  readily  be  perceived  that  as  the  force  varies  inversely 
as  the  distance,  and  the  motion  also,  that  those  two  elements 
preserve  the  same  proportions  to  each  other  throughout  the  orbit, 
namely,  the  same  in  any  other  part  of  the  orbit  that  they  do  in 
the  mean  points  of  force  and  motion. 

And  as  the  rate  of  convergency  is  inversely  as  the  period, 
hence,  the  rate  of  convergency  depends  upon  the  velocity  in  one 
and  the  same  orbit ;  so  that  if  the  velocity  be  double  in  one 
point  of  an  orbit,  in  which  the  curvature  is  equal  to  what  it  is  in 
another  point  of  the  orbit,  the  rate  of  convergency  will  also  be 
double,  &c. 

Hence  then,  those  elements,  namely,  motion,  force,  and  con- 
vergency, necessarily  and  consequently  preserve  the  same  ratio 
to  each  other  throughout  the  orbit,  that  they  do  in  the  mean 
points ;  and  hence  the  rate  of  force,  or  gravity,  is  a  mean  pro- 
portional between  the  rate  of  motion  and  the  rate  of  convergency, 
throughout  the  entire  orbit ;  nor  are  they  under  any  necessity, 
as  will  be  clearly  perceived,  of  alternately  getting  the  better  of 
each  other,  in  order  to  keep  up  the  balance  of  power. 

Thus  the  mean  motion  is  inversely  as  the  square  of  the  mean 
distance ;  and  the  actual  motion  varies  from  the  point  of  mean 
motion  inversely  as  the  distance  varies ;  and  hence,  we  deduce 


ON    PLANETARY    FORCE    AND    MOTION.  211 

this  general  rule  for  finding  the  rate  of  motion  at  any  given  dis- 
tance of  the  planet  in  its  orbit  from  the  sun,  namely  : 

The  product  of  the  mean  distance ,  by  the  rate  of  mean  motion 
divided  by  the  actual  distance  at  any  given  point  of  the  orbit, 
gives  the  rate  of  motion  at  such  given  point. 

And  upon  no  other  principle  can  the  motion  vary  inversely  as 
the  distance  varies  ;  nor  the  planet  describe  equal  areas  in  equal 
times,  nor  the  square  of  the  period  be  the  cube  of  the  distance. 
And  thus,  gravity  governs  as  perfectly  throughout  the  orbit  as 
though  it  were  a  centric  orbit.  There  is  no  more  necessity  of 
calling  to  aid  a  centrifugal  force,  clothed  with  such  powers  as  are 
found  necessary  to  keep  the  balance  of  power;  no  more  necessi- 
ty of  resorting  to  the  popular  method  of  accounting,  by  alleging 
that  motion  and  force  alternately  get  the  better  of  each  other,  than 
if  the  orbit  were  centric ;  for  the  planet  is  as  obedient  to  the 
force  of  gravity,  and  sings  through  the  heaven's  with  the  same 
harmony  and  concord  in  the  eccentric,  as  in  the  centric  orbit. 

Thus,  in  any  eccentric  orbit,  the  motion  of  the  planet  at  any 
given  point  of  the  orbit  varies  inversely  as  the  distance  varies  ; 
hence,  the  motion  at  any  given  point  of  the  orbit,  will  be  the 
square  of  what  the  motion  would  be  in  a  corresponding  point  of 
a  like  orbit  to  that  of  the  given  orbit,  whose  mean  distance  is 
the  square  root  of  the  mean  distance  of  the  given  orbit;  be- 
cause the  mean  motion  of  a  planet  is  always  inversely  as  the 
square  root  of  the  mean  distance. 

So  if  the  mean  distance  of  a  planet  be  4,  and  its  perihelion 
distance  be  1,  the  actual  motion  at  any  given  point  of  the  orbit, 
will  be  the  square  of  what  the  actual  motion  of  a  planet  revolv- 
ing in  a  centric  orbit  would  be,  if  revolving  at  just  half  the 
distance  of  such  given  point. 

Thus  if  the  mean  distance  be  4,  the  motion  at  the  mean  dis- 
tance will  be  .5,  —  or  the  square  of  the  motion  of  a  planet  revolv- 
ing in  a  centric  orbit,  about  the  same  central  force,  at  the  distance 
of  2.  And  if  the  mean  distance  be  4,  and  the  perihelion  distance 
be  1,  the  motion  at  the  perihelion  will  be  2,  or  the  square  of  the 
motion  of  a  planet  revolving  in  a  centric  orbit  about  the  same 
central  force  at  the  distance  .5. 

Let  us  now  suppose  a  case  or  two  :  and  firstly,  suppose  a 
planet  revolving  in  an  orbit  whose  mean  distance  is  4,  and  whose 
perihelion  distance  is  1.  In  such  case  we  know  the  rate  of 
mean  motion  is  .5 ;  the  rate  of  mean  force  .25  ;  and  the  rate  of 
mean  convergency  .125  ;  and  the  rates  of  motion,  force  and  con- 
vergency,  each,  become  quadrupled  at  perihelion,  or  when  the 
distance  has  become  l-4th  of  the  mean  distance ;  that  is,  when  the 
planet  has  arrived  at  perihelion,  the  rate  of  motion  will  be  2,  the 
rate  of  force  1,  and  the  rate  of  convergency  .5 ;  thus  retaining 


212  ON    PLANETARY    FORCE    AND    MOTION. 

the  same  proportion  to  each  other,  and  the  same  ability  to  per- 
form their  journey,  or  pursue  the  line  of  the  orbit,  as  at  any 
other  point  of  the  orbit. 

Thus  we  perceive  that  the  force  from  aphelion  to  perihelion, 
is  constantly  dividing  its  accumulating  power  between  the  mo- 
tion and  the  convergency,  according  to  their  necessities  ;  and 
from  perihelion  to  aphelion,  the  force  restrains  and  diminishes  their 
rates  in  like  proportions  to  that  of  their  increase  from  aphelion  to 
perihelion  ;  and  in  such  case  the  motion  becomes  accelerated 
and  retarded,  or  rather  increased  and  diminished,  in  the  same 
proportion  that  the  convergency  becomes  increased  and  dimin- 
ished ;  or  inversely  as  the  distance  is  increased  or  diminished. 
Hence,  the  motion  is  not  accelerated  by  the  same  law  of  pro- 
gression, in  respect  to  time  and  force,  that  a  falling  body  near 
the  earth  is  ;  in  whiqh  the  progression  of  acceleration  is  constant- 
ly as  the  amount  of  the  next  succeeding  odd  number  is  to 
unity. 

Thus,  when  the  foregoing  planet  arrives  at  perihelion,  the 
force  is  the  same  as  would  be  required  for  it  to  revolve  in  a 
centric  orbit  at  that  distance ;  but  the  rate  of  motion  having 
become  double  what  would  be  required  for  a  planet  to  revolve 
in  a  centric  orbit  at  that  distance,  the  rate  of  convergency  is 
consequently  one  half  of  what  would  be  required ;  and  hence, 
the  planet  pursues  the  curvature  of  the  orbit  dictated  by  the 
mean  distance. 

In  the  foregoing  we  perceive  that  the  general  rule  applies, 
namely,  the  product  of  the  mean  distance,  by  the  mean  rate  of 
motion,  divided  by  the  actual  distance  of  any  point  of  the  orbit, 
gives  the  rate  of  motion  at  such  point.  So  also  this  general 
rule  is  equally  applicable  to  the  rate  of  convergency,  namely,  the 
product  of  the  mean  distance  by  the  mean  rate  of  convergency, 
divided  by  the  actual  distance  at  any  given  point  of  the  orbit, 
gives  the  rate  of  convergency  at  such  given  point ;  and  this  gen- 
eral rule  also  equally  applies  to  the  force. 

This  then  is  the  simple  and  easy  construction  of  the  laws  of 
force,  motion,  and  convergency,  in  an  eccentric  orbit,  based  upon 
the  hypothesis  that  force  is  inversely  as  the  distance ;  but  if  we  go 
upon  the  hypothesis,  that  gravity  is  inversely  as  the  square  of  the 
distance,  and  append  to  it  that  extraordinary  though  prevalent 
version  of  Kepler's  other  law,  namely,  that  the  motion  is  inversely 
as  the  distance,  (and  without  which  appendage,  even  the  idea  or 
notion,  that  force  is  inversely  as  the  square  of  the  distance,  could 
not  have  been  sustained  for  a  moment,)  and  take  the  foregoing 
planet  whose  mean  distance  is  4,  and  whose  perihelion  distance 
is  1,  we  shall  find  among  others,  the  following  results,  namely : 

The  period  must  be  16,  and  the  mean  rate  of  force  .0625,  or 


ON    PLANETARY    FORCE    AND    MOTION.  213 

the  inverse  of  the  period,  and  equal  to  the  mean  rate  of  conver- 
gency ;  and  the  mean  rate  of  motion  must  be  .25,  or  inversely 
as  the  mean  distance. 

When  the  planet  has  arrived  at  the  perihelion,  the  force  will 
be  1,  or  16  times  its  mean ;  and  the  motion  will  be  1,  or  four 
times  its  mean ;  and  what  should  hinder  the  planet  from  then 
revolving  in  a  centric  orbit  at  that  distance,  may  not  readily  be 
perceived  ;  as  the  force  and  motion  both  conspire  to  that  effect ; 
there  being  no  extra  motion  to  cause  a  less  rate  of  convergency 
than  that  of  the  force  or  motion ;  for  it  would  be  futile  to  say  or 
assume  that  the  rate  of  convergency  was  any  other  than  unity, 
when  there  was  no  cause  for  it. 

But  it  may  be  sard  that  the  actual  period  is  8,  the  mean  rate 
of  motion  .5,  and  the  mean  rate  of  convergency  125,  and  that 
those  elements  will  always  have  the  same  proportions,  whatever 
be  the  law  of  force  ;  and  hence,  at  perihelion,  or  distance  1,  all 
the  elements  would  have  the  same  ratio  to  the  rate  of  force, 
whatever  be  its  laws ;  that  in  such  case,  the  rate  of  force  at  per- 
ihelion would  be  a  mean  proportional  between  the  rate  of 
motion  and  the  rate  of  convergency. 

This  mean  proportional,  it  will  be  perceived,  could  only  hap- 
pen at  the  point  of  perihelion,  and  that  too  when  at  the  distance 
1.  But  in  all  Bother  parts  of  the  orbit,  while  the  rates  of  motion 
and  convergency  preserved  the  same  proportions  to  each  other, 
and  varied  inversely  as  the  distance,  the  force  would  be  wander- 
ing, with  little  apparent  regard  to  its  change,  over  the  motion  and 
convergency. 

But  I  have  already  fully  shown  that  the  only  condition  upon 
which  the  force  can  be  inversely  as  the  square  of  the  distance, 
would  be  that  the  mean  motion  of  a  planet  should  be  inversely  as 
the  mean  distance,  instead  of  being  what  it  is,  namely,  inversely 
as  the  square  of  the  distance;  than  which,  a  more  plain  or 
palpable  fact  could  not  well  be  desired. 

I  will  now  attempt  to  make  some  few  simple  applica- 
tions of  my  determinations  of  the  measures  of  the  circle, 
to  the  phenomena  of  planetary  motion,  as  corroborated  by 
the  great  Keplerian  law,  and  the  eternal  fitness  of  the  econ- 
omy of  numbers  for  the  development  of  the  laws  of  nature,  to 
the  human  understanding;  —  in  which  I  will  insist  that  the  great 
Architect  of  the  Universe  had  Unity  of  Purpose  in  view,  when 
he  created  and  established  it  by  everlasting  laws,  which  are  as 
necessarily  the  laws  of  order,  regularity,  and  the  eternal  fitness 
of  things,  as  they  are  the  laws  of  power  and  supremacy; 
otherwise  we  should  discover  but  little  of  the  attribute  of  wis- 
dom; and  hence,  truth,  when  once  discovered,  may  at  least, 
be  as  beautiful  as  error. 


214  ON  PLANETARY  FORCE  AND  MOTION. 

I  will  here  remark,  that  the  popular  measures  of  the  circle 
apply  to  the  great  Keplerian  law,  that  the  squares  of  the  periods 
of  a  system  of  planets,  are  as  the  cubes  of  their  mean  distances 
from  the  sun,  in  the  same  manner  as  (and  no  better  than)  they 
apply  to  the  measures  of  solids  composed  of  plane  surfaces. 
Hence,  in  applying  the  popular  measures  of  the  circle  to  the  law 
by  which  a  system  of  planets  revolve  in  their  orbits,  as  well  as 
in  the  application  of  the  circle  to  the  measure  of  plain  solids, 
it  will  always  be  found  necessary  in  order  to  equate  or  finish  out 
the  work,  to  have  on  hand  a  sufficient  number  of  minim  quanti- 
ties, as  a  kind  of  calculi  to  chink  up  with ;  and  that  too,  in 
proportion  as  the  extension  of  the  work  may  serve  to  place  it 
out  of  square. 

I  am  riot  going  to  assume  that  the  infinity  of  coincidences  and 
reciprocities  which  continually  flow  from  a  comparison  of  my 
measures  of  the  circle  with  measures  of  plain  solids,  or  with  the 
laws  of  planetary  motion,  as  determined  by  Kepler's  laws,  is  by 
any  means  to  supply  the  place  of  absolute  proof  of  the  correctness 
of  my  determination ;  but  such  must  depend  upon  the  re- 
sults and  absolute  conclusions  drawn  from  an  inductive  course 
of  geometric  reasoning ;  in  which  one  portion  of  the  task  will 
be  to  exhibit  and  show  the  errors  of  the  popular  methods,  and 
thereby  to  endeavor  to  unloose  that  implicit  faith  in  those  errors 
which  have  become  chained  and  riveted  just  in*  proportion  as 
the  inquirer  has  examined  or  attended  to  the  subject;  and  for 
the  accomplishment  of  which,  it  will  be  necessary  in  the  consid- 
eration of  the  quadrature,  to  adopt  certain  appropriate  (but 
very  simple)  signs,  characters  or  symbols,  which  cannot  here  be 
used  to  advantage. 

I  will  therefore  simplify  the  present  explanation  as  far  as  pos- 
sible, —  and  will  call  l-4th  of  the  circumference  of  an  orbit,  the 
quadrant  of  the  orbit;  l-8th  of  the  circumference,  the  octant  of  the 
orbit,  &c.  Radius  of  an  orbit  of  course  denotes  one  half  of  the 
diameter  of  the  orbit. 

I  will  notice  that  what  is  properly  the  prime  orbit,  is  that 
whose  radius  is  unity  or  1,  having  its  diameter  2,  for  the  reason, 
that  the  squares  of  the  periods  of  a  system  of  planets,  are  as 
the  cubes  of  their  mean  distances  from  the  sun  ;  and  consequent- 
ly when  the  distances  of  the  planets  of  a  system  are  numerically 
proportioned  to  any  planet  of  the  system  whose  distance  is 
assumed  at  unity ;  the  square  of  the  period  of  each  planet  of 
the  system  will  be  the  cube  of  its  own  distance ;  consequently, 
when  the  distance  of  a  planet  is  assumed  at  unity  or  1,  its  period 
is  necessarily  1.  Nevertheless,  what  is  properly  the  prime  circle, 
is  that  whose  diameter  is  unity,  having  its  radius  but  half  of 
unity.  Such  circle  will  be  found  to  afford  vastly  more  facili- 


ON    PLANETARY    FORCE    AND    MOTION.  215 

ties  for  deducing  the  true  quadrature,  than  any  other  numerical 
dimension. 

But  to  proceed  with  a  short  application  of  my  measures  of 
the  circle  to  planetary  motion. 

1.  By  the  Keplerian  law,  the  square  of  the  period  is  the  cube 
of  the  distance  ;  or  in  other  words,  the  period  of  a  planet  is  the 
square  root  of  the  third  power  of  the  mean  distance.  And 
according  to  my  measures  of  the  circle,  when  the  quadrant  of 
an  orbit  is  4,  (or  the  circumference  16,)  the  square  of  the  quad- 
rant of  the  orbit  is  the  cube  of  radius.  And  this  we  know  from 
Kepler's  law,  that  when  the  square  of  the  quadrant  is  the  third 
power,  or  cube  of  radius,  the  quadrant  of  the  orbit  is  conse- 
quently numerically  equal  to  the  period. 

But  by  the  popular  measures  of  the  circle,  in  order  that  the 
period  may  be  equal  to  the  quadrant  of  the  orbit,  the  quadrant 
must  be  very  little  less  than  4,  or  the  circumference  of  the  orbit 
a  very  little  less  than  16. 

By  Kepler's  law  the  period  of  a  planet  is  always  the  square 
root  of  the  third  power,  or  cube  of  radius ;  and  by  my  determi- 
nation of  the  circle,  the  quadrant  of  any  orbit  is  numerically 
equal  to  the  third  root  of  the  square  of  twice  the  period.  Hence, 
when  the  quadrant  of  an  orbit  is  1,  radius  is  the  third  root  of 
.25,  and  the  period  will  be  the  second  or  square  root  of  .25,  or 
equal  to  half  of  the  quadrant,  or  to  l-8th  of  the  circumference  of 
the  orbit ;  and  we  know  that  when  the  diameter  of  an  orbit  is  8, 
the  period  is  equal  to  diameter,  or  to  twice  radius;  for  then  the 
third  power  of  radius  will  be  the  square  of  the  diameter  as  well 
as  of  the  period.  So  by  my  determination,  when  the  quadrant  of 
an  orbit  is  2,  the  period  of  the  planet  will  be  the  square  root  of  2, 
and  radius  of  the  orbit  will  be  the  cube  root  of  2.  And  when  the 
diameter  of  an  orbit  is  2,  (namely,  when  the  orbit  is  prime,)  the 
period  is  necessarily  equal  to  radius. 

So  when  the  diameter  of  the  orbit  is  4,  the  period  will  be  the 
square  root,  and  radius  the  cube  root  of  8,  —  for  it  will  be  recol- 
lected that  the  period  is  always  the  square  root  of  the  same 
numerical  quantity  of  which  radius  is  the  cube  root. 

Again,  by  my  determination,  four  times  the  square  of  the 
period  of  a  planet,  is  always  the  third  power  of  the  quadrant  of 
the  orbit ;  and  radius  of  any  orbit  is  always  the  farther  of  two 
mean  proportionals  from  unity  to  the  time  of  the  period  ;  for 
radius  is  always  the  third  root  of  the  square  of  the  period. 

So  by  my  determination,  when  the  period  of  a  planet  is  .5  or 
half  of  unity,  it  is  equal  to  half  the  quadrant  of  the  orbit. 
When  the  period  is  4,  the  period  is  equal  to  the  quadrant  of  the 
orbit.  When  the  period  is  32,  it  is  equal  to  twice  the  quadrant 
of  the  orbit ;  and  when  the  period  is  256,  (or  square  of  16,)  the 


216  ON    PLANETARY    FORCE    AND    MOTION. 

period  is  equal  to  four  times  the  quadrant,  or  is  equal  to  the 
circumference. 

When  the  diameter  of  an  orbit  is  8,  the  period  is  equal  to  the 
diameter ;  and  when  the  quadrant  of  an  orbit  is  4,  the  period  is 
equal  to  the  quadrant. 

By  the  first  allegation,  the  square  of  the  diameter  or  the  third 
power  of  the  radius  ;  and  by  the  second  proposition,  the  square 
of  the  quadrant,  is  equal  to  the  third  power  of  the  radius. 

To  the  first  proposition,  the  world  have  assented,  because  it  is 
plain  to  see,  that  the  square  of  8  is  the  third  power  of  4.  But 
in  respect  to  the  last  proposition,  by  the  popular  measures  of  the 
circle,  when  the  quadrant  of  an  orbit  is  4,  the  period  is 
4.063750,  thus  requiring  a  minim  ratio  of  .063750  to  equate  the 
work  with. 

By  my  measures  of  the  circle,  the  square  of  the  period  when 
the  diameter  of  the  orbit  is  the  square  root  of  64,  is  the  third 
power  of  the  period,  when  the  quadrant  of  the  orbit  is  the 
third  root  of  64.  Or  thus,  the  square  of  the  period,  when  radius 
of  the  orbit  is  4,  is  the  third  power  of  the  period  when  the 
quadrant  of  the  orbit  is  4. 

But  by  the  popular  measures  of  the  circle,  the  square  of  the 
period  when  radius  of  the  orbit  is  4,  would  be  the  third  power 
of  the  period  when  the  quadrant  of  the  orbit  is  a  little  less  than 
4,  namely,  3.978988. 

I  will  now  just  refer  to  my  prime  table,  upon  which  I  propose  to 
explain  many  of  the  principles  of  the  quadrature  of  the  circle,  for 
the  purpose  of  showing  its  applicability  to  the  laws  of  planetary 
motion  also.  Hence,  if  we  take  any  term  of  the  table  for  the 
mean  distance  of  a  planet,  we  readily  obtain  in  other  terms  of 
the  table,  the  period,  the  rate  of  mean  motion,  the  mean  rate  of 
force,  and  the  mean  rate  of  convergency  of  the  planet,  —  as  also 
the  whole  amount  of  motion,  of  force,  and  of  convergency  during 
the  period  of  the  planet;  the  amount  of  force  during  an  entire 
period  being  always  a  mean  proportional  between  the  amount 
of  convergency,  and  the  amount  of  motion  during  the  period. 

Thus,  if  the  mean  distance  of  a  planet  be  the  third  root  of  4, 
for  instance,  the  time  of  the  period  will  be  the  third  or  cube  root 
of  8,  the  rate  of  mean  motion  will  be  the  third  root  of  .5,  the 
rate  of  mean  force  will  be  the  third  root  of  .25,  and  the  rate  of 
mean  convergency  will  be  the  third  root  of  .125.  The  whole 
amount  of  convergency  during  the  period  will  be  1,  the  whole 
amount  of  force  during  the  period  will  be  the  third  root  of  2, 
and  the  whole  amount  of  motion  during  the  period  will  be  the 
third  root  of  4. 


CHAPTER   IV. 


On  the  Hypothesis  suggested  by  Kepler  in  respect  to  Elliptical 

Orbits. 

PERHAPS  one  reason  that  may  be  offered  why  it  is  not  as  im- 
perious upon  philosophy  and  astronomy  that  the  elliptical  hy- 
pothesis of  Kepler  should  be  adopted,  as  that  his  other  two 
laws  should,  is,  that  any  general  law  of  gravity  or  motion,  ap- 
plies equally  well,  at  least  to  an  eccentric  circular  orbit,  as  to  the 
ellipse. 

Thus  Newton's  famous  demonstration  of  Kopler's  hypothesis, 
that  the  motion  of  a  planet  revolving  in  an  eccentric  orbit  varies 
inversely  as  the  distance  varies,  and  hence  that  the  radius  vec- 
tor will  pass  over  or  describe  equal  areas  in  equal  times,  by  an 
application  of  the  problem,  "  That  triangles  on  the  same  base, 
or  on  equal  bases,  having  the  same  height  are  equal  to  each 
other,"  is  equally  applicable  to  a  circular  orbit,  whether  centric 
or  eccentric,  as  to  the  ellipse ;  nay  further,  it  applies  equally 
well  to  the  parabola,  or  hyperbola. 

Hence  there  appears  to  be  no  necessity  that  a  planet  should 
revolve  in  an  ellipse  with  the  sun  in  one  of  the  foci,  in  conse- 
quence of  such  demonstration  ;  as  it  can  equally  well  revolve  in 
conformity  with  such  demonstration,  in  a  circle,  having  the  sun 
stationary  in  any  given  point  within  the  circumference  of  the 
orbit. 

Nevertheless,  it  is  not  so  much  my  intention  to  establish  the 
particular  forms  of  orbits,  as  to  promulgate  the  true  laws  of 
gravity  and  motion ;  to  endeavor  to  reconcile  their  complicated 
simplicity,  and  to  remove  that  blur  which  has  so  long  perplexed 
the  student  and  inquirer  when  endeavoring  to  get  an  insight 
into  the  laws  of  force  and  motion  through  the  medium  of  those 
round  and  round  explanations  given  by  Maclaurin,  Ferguson, 
and  others  ;  by  which  we  are  given  to  understand,  that  the  ve- 
locity sometimes  gets  the  better  of  the  force,  and  sometimes 
the  force  gets  the  better  of  the  velocity,  and  that  it  has  been 
found  necessary,  in  order  to  regulate  or  equate  the  defect  in  the 
28 


218  ON  KEPLER'S  HYPOTHESIS 

eternal  laws,  to  institute,  as  a  kind  of  balance  wheel,  what  they 
term  the  centrifugal  force,  which,  by  the  aid   of  such  forces  as 
.  they  choose  to  bestow  upon  it,  enables  them  to  make  the  requi- 
site equations. 

Mr.  Norton,  in  his  Astronomy,  gives  the  following  version  of 
what  are  called  Kepler's  three  laws,  viz : 

1.  The  areas  described  by  the  radius  vector  of  a  planet  are 
proportional  to  the  times. 

2.  The  orbit  of  a  planet  is  an  ellipse,  of  which  the  sun  oc- 
cupies one  of  the  foci. 

3.  The  squares  of  the  times  of  revolution  of  the  planets   are 
proportional  to  the  cubes  of  their  mean  distances  from  the  sun, 
or  of  the  semi-major  axes  of  their  orbits. 

And  Mr.  Norton  says  that  "  The  first  two  Kepler  assumed  as 
hypotheses,  (on  partial  examination,)  after  he  had  discovered 
that  the  radius  vector  and  angular  motion  of  a  planet  were 
equable,  and  afterwards  verified  them,  or  partially  verified  them, 
—  and  they  have  been  completely  verified  by  other  astrono- 
mers,"—  and  promises  to  show  in  the  sequel,  that  they  are  ver- 
ified by  the  results  deducible  from  them  ;  and  remarks,  "  that 
these  two  laws  being  established,  the  third  is  obtained  by  simply 
comparing  the  known  major  axes  and  times  of  revolution,"  and 
then  alleges,  that  "  the  apparent  motion  of  the  sun  in  space  must 
be  subject  to  Kepler's  laws." 

That  the  apparent  motion  of  the  sun  in  space  is  subject  to 
Kepler's  first  and  third  laws,  there  can  be*ho  doubt.  But  that  it 
is  subject  to  the  second,  must  depend  upon  the  proof  of  the  el- 
liptical hypothesis ;  and  that  must,  probably,  be  deduced  from 
observation  alone,  if  proved  at  all ;  as  no  physical  cause  appears 
to  come  forth  in  support  of  it. 

Mr.  Norton  says,  "  The  orbits  of  the  sun,  moon  and  planets 
being  regarded  as  ellipses,  the  perigee  and  apogee,  or  the  per- 
ihelion and  aphelion,  are  the  extremities  of  the  major  axis  of  the 
orbit." 

This  is  true,  if  we  regard  the  orbits  as  ellipses  ;  for  the  ellipse 
is  what  presents  a  major  axis,  the  extremities  of  which  must  be 
the  apogee  and  perigee.  But  it  is  the  eccentricity  of  the  orbit 
which  gives  the  apogee  and  perigee,  and  this  whether  the  orbit 
be  an  ellipse  or  a  circle. 

Mr.  Norton  again  says,  "  The  law  of  angular  motion  of  a 
planet  about  the  sun,  may  be  deduced  from  Kepler's  first  law." 

This  with  proper  qualifications  is  doubtless  true ;  but  it  has 
nothing  whatever  to  do  with  the  elliptical  hypothesis,  as  it 
depends  wholly  upon  the  eccentricity  of  the  orbit,  and  not  upon 
the  ellipticity ;  for  when  the  orbit  is  eccentric,  what  astronomers 


RESPECTING    ELLIPTICAL    ORBITS.  219 

call  the  angular  motion,  must  of  necessity  exist.  And  if  the 
actual  motion  varies  inversely  as  the  distance  varies,  the  radius 
vector  must  of  necessity  describe  equal  areas  in  equal  times,  — 
whatever  be  the  form  of  the  orbit ;  and  this  in  accordance  with 
Newton's  said  demonstration,  from  which  so  much  Astronomical 
calculation  has  been  drawn ;  as  that  bodies  may  form  their 
perihelions  in  parabolic  or  hyperbolic  curves  ;  and  may  revolve 
about  the  sun  in  all  kinds  of  ellipses  however  eccentric. 

Nor  do  I  think  that  Mr.  Norton's  explanations,  given  in  Sec- 
tions 184  and  185  of  his  Astronomy,  have  improved  at  all  upon 
Newton's  demonstration,  or  that  they  in  anywise  prove  that  an 
eccentric  orbit  must  of  necessity  be  an  ellipse.  Nor  does  he  even 
pretend  to  deduce  the  angular  motion  from  any  other  cause  than 
that  of  the  eccentricity  of  the  orbit ;  for  he  only  assumes  or 
considers  the  orbit  used  by  way  of  explanation,  to  be  an  ellipse. 

Thus  far  it  would  seem  as  though  nothing  had  been  shown, 
proved,  or  demonstrated,  in  favor  of  the  ellipse,  that  does  not 
equally  apply  to  the  eccentric  circular  orbit ;  and  that  the  ellip- 
tical hypothesis  must  still  rest  for  its  support  upon  evidence  to 
be  wholly  drawn  from  observation,  (if  supported  at  all,)  in  lieu 
of  being  drawn  from  any  physical  cause  that  can  be  assigned. 

But  so  general  is  the  belief  in  the  elliptical  hypothesis,  that 
there  remains  perhaps  as  little  apprehension  that  the  orbits  are 
not  ellipses  at  the  present  day,  as  that  they  were  so  previous  to 
the  days  of  Kepler.  And  hence  we  find  the  constant  attempt 
in  all  astronomical  works,  by  almost  every  mode  of  expression 
that  can  be  devised,  to  induce  a  belief  in  the  elliptical  hypoth- 
esis ;  and  often,  too,  in  a  way  and  manner  that  would  seem  not 
to  be  required  for  the  purposes  of  explanation  of  the  subject 
directly  under  consideration,  —  as  though  it  were  an  imperious 
point  of  faith,  however  reluctant  the  mind  should  be  to  adopt 
it. 

Thus  Mr.  Norton  says,  "  The  eccentricity  of  an  elliptic  orbit, 
is  the  distance  between  the  centre  of  the  orbit  and  either  focus ; " 
and  if  a  circular  orbit  be  eccentric,  the  eccentricity  is  the  distance 
between  the  centre  of  the  orbit  and  the  sun ;  the  same  as  in  the 
ellipse. 

Again  he  says,  "  The  most  accurate  method  of  determining 
the  longitude  and  epoch  of  the  perigee,  rests  upon  the  principle 
that  the  apogee  and  perigee  are  the  only  two  points  of  the  orbit 
whose  longitudes  differ  by  180°,  in  passing  from  one  to  the  other 
of  which  the  sun  employs  just  half  of  the  year.  This  prin- 
ciple may  be  inferred  from  Kepler's  law  of  areas ;  for  it  is  a 
well  known  property  of  the  ellipse,  that  the  major  axis  is  the 
<only  line  drawn  through  the  focus,  that  divides  the  ellipse  into 


220  ON  KEPLER'S    HYPOTHESIS 

equal  parts,  and  by  the  law  in  question  equal  areas  correspond 
to  equal  times." 

Now  the  manner  in  which  this  sentence  is  constructed,  has 
somewhat  the  appearance  of  a  design  to  sustain  a  favorite  theory 
which  is  lacking  in  the  requisite  proof,  and  hence  must  be  sus- 
tained by  presumptive  evidence  however  futile. 

Now  not  a  single  important  fact  alleged  in  the  sentence  arises 
from  the  properties  of  the  ellipse ;  for  the  truth  is,  that  in  either 
an  ellipse  or  a  circular  orbit  having  eccentricity,  the  apogee  and 
perigee  are  the  only  points  of  the  orbit  whose  longitudes  differ 
by  180°  in  passing  from  one  to  the  other,  of  which  the  sun  em- 
ploys just  half  the  year;  and  that  the  line  of  the  apsides  is  the 
only  line  that  can  pass  through  the  sun  and  divide  the  orbit  into 
equal  parts  at  the  same  time,  is  as  true  of  the  eccentric  circle  as 
of  the  ellipse.  And  the  proof  arising  from  the  application  of 
Euclid's  problem  in  support  of  Kepler's  first  law,  —  viz,  "  That  if 
a  body  be  made  to  revolve  around  a  fixed  point  with  a  motion 
varying  inversely  as  the  distance,  a  line  drawn  from  such  fixed 
point  to  such  revolving  body,  will  describe,  or  pass  over,  areas 
proportioned  to  the  times,"  —  is  as  true  in  respect  to  the  circle  as 
to  the  ellipse. 

Having  consulted  Mr.  Norton  thus  far,  I  am  unable  to  dis- 
cover that  he  has  furnished  the  slightest  evidence  in  favor  of  the 
elliptical  hypothesis;  notwithstanding  we  find  from  his  preface, 
that,  in  his  compilation,  he  has  consulted  the  most  eminent  au- 
thors upon  astronomy  and  mathematics  ;  and  hence,  thus  far 
physical  astronomy  does  not  seem  to  furnish  equal  evidence  in 
support  of  the  elliptical  hypothesis  that  it  does  in  favor  of  Kep- 
ler's other  laws.  And  hence  the  evidence  in  support  of  said  hy- 
pothesis seems  rather  to  be  the  result  of  Kepler's  observations, 
to  which  an  easy  or  ready  assent  has  been  given  ;  and  especially 
since  the  celebrated  demonstration  in  respect  to  equal  areas  in 
equal  times,  which  seemed  to  operate  as  a  charm,  even  upon 
matters  with  which  it  had  no  concern. 

But  it  is  possible  that  the  consequences  resulting  from  the 
adoption  by  Kepler,  of  the  elliptical  hypothesis,  in  connection 
with  Newton's  demonstration  of  Kepler's  first  law,  and  his 
especial  application  of  it  to  Kepler's  elliptical  hypothesis,  are  not 
such  at  the  present  day,  in  physical  astronomy,  as  a  better  dis- 
pensation would  have  produced;  for  the  problem  applying 
equally  well  to  an  ellipse  of  extreme  eccentricity  as  to  any  oth- 
er, neither  Newton,  Halley,  nor  the  world,  have  set  any  bounds 
to  the  ellipticity  of  their  orbits.  For  although  both  Newton  and 
Halley,  previous  to  the  supposed  important  demonstration,  sup- 
posed the  comets  to  perform  their  perihelions  in  parabolic  curves, 


RESPECTING    ELLIPTICAL    ORBITS.  221 

they  changed  their  views  after  such  demonstration,  even  in  re- 
spect to  cometary  motion. 

They  hence  allow  575  years  for  the  period  of  the  comet  which 
appeared  in  1680,  and  give  it  an  elliptical  orbit  so  eccentric  that 
its  curvature  at  its  perihelion  must  be  very  rapid  indeed,  it  being 
166  times  nearer  the  sun  than  the  earth  is.  And  notwithstanding 
the  astonishing  velocity  with  which  it  is  said  lo  move  in  its 
perihelion,  yet  if  the  orbit  were  an  ellipse  with  the  sun  in  one 
of  the  foci,  the  ellipticity  would  be  so  great  or  extreme,  that  if 
another  comet  were  to  revolve  in  a  circle  with  its  line  of  apsides 
of  equal  length  and  of  equal  eccentricity,  (and  Mr.  Norton  says 
that  the  line  of  the  apsides  is  what  determines  the  length  of  the 
period,)  the  motion  of  the  comet  revolving  in  the  ellipse  would  be 
but  about  2-3ds  as  great  as  that  of  the  comet  revolving  in  the  cir- 
cle, in  any  corresponding  part  of  the  orbit,  —  as  will  be  self-evident 
to  any  one  who  wTill  examine  the  subject  but  for  a  moment ; 
while  the  amount  of  the  force  of  gravity  applied  to  the  comet 
revolving  in  the  circle  in  the  time  of  its  period,  would,  upon  any 
hypothesis  of  the  law  of  gravity,  be  vastly  less  than  the  amount 
applied  to  the  comet  revolving  in  the  ellipse  in  the  time  of  its 
period,  or  in  the  same  time. 

Bat  this  part  of  the  subject  belongs  more  particularly  to  the 
laws  of  gravity  and  motion. 

But  notwithstanding  the  elliptical  hypothesis  is  such  a  favorite 
in  the  science  of  astronomy,  it  is  somewhat  surprising  that  it 
should  be  so,  seeing  its  application  is  so  repugnant  to,  and  irre- 
concilable with,  the  laws  of  gravity  and  motion  ;  for  though  it  is 
alleged  that  the  orbits  of  some  of  the  comets  are  known  from 
observation  to  be  very  eccentric  ellipses,  yet  there  are  but  three, 
namely,  Encke's,  Biela's,  and  Halley's,  whose  periods  and  orbits 
have  been  determined. 

But  it  is  alleged  by  Mr.  Vince  and  others,  that  astronomers  do 
not  calculate  the  elements  of  a  visible  comet  upon  the  princi- 
ple that  the  orbit  is  an  ellipse,  but  upon  the  principle  that  it  is  a 
parabola,  and  afterwards  deduce  the  elliptical  form  from  those 
calculations. 

The  elliptical  hypothesis,  then,  seems  to  receive  but  little,  if 
any  aid,  by  its  application  to  cometary  motion,  it  being  sus- 
tained neither  by  any  known  physical  laws,  nor  by  observation 
upon  which  dependence  can  be  placed. 

Let  us,  then,  recur  to  what  Mr.  Vince  says  in  respect  to  Kep- 
ler's investigations  of  the  subject,  and  which  induced  him  to 
adopt  and  promulgate  the  hypothesis. 

Ptolemy  supposed  that  the  orbits  of  the  planets  were  circles, 
and  that  the  earth  was  not  in  the  centre  of  those  orbits,  but  situ- 


222 

ated  at  some  distance  from  the  centre  of  those  orbits,  around 
which  the  planets  revolved  with  an  equable  motion  ;  and  thus 
he  endeavored  to  account  for  the  apparent  inequality  of  a  planet's 
motion,  requiring  therefore  an  equation  of  its  orbit. 

Tycho  adopted  this  theory  in  part,  retaining  the  theory  of  the 
equable  motion  of  a  planet  around  a  centre,  but  placed  that  cen- 
tre at  a  different  distance  from  the  earth  from  that  at  which  Ptolemy 
placed  it,  by  which  his  observations  and  computations  would 
agree  within  a  few  minutes;  but  as  the  eccentricity,  calculated 
from  the  equation,  would  not  agree  with  his  supposition,  he  con- 
cluded that  the  sun  was  not  always  at  the  same  distance  from  the 
central  point,  which  made  Kepler  suspect  that  the  centre  was  not 
the  point  about  which  the  motion  was  equal. 

Hence  Kepler  instituted  a  series  of  calculations  upon  the  orbit 
of  Mars,  based  mostly  upon  Tycho's  observations,  for  the  pur- 
pose of  ascertaining  its  eccentricity  upon  the  assumption  that  it 
was  a  circle ;  and  calling  the  distance  of  the  earth  100000,  he 
found  the  aphelion  distance  of  Mars  to  be  166780,  and  the  peri- 
helion distance  138500,  and  hence  its  mean  distance  152640, 
and  its  eccentricity  14140. 

He  next  determined  three  other  distances  of  Mars,  and  found 
them  to  be  147750,  163100,  and  166255 ;  and  on  calculating 
their  distances  upon  the  supposition  that  the  orbit  was  a  circle, 
he  found  that  they  would  corne  out  148539, 163883,  and  166605, 
and  that  the  errors,  upon  the  hypothesis  that  the  orbit  was  a  cir- 
cle, would  be  789,  783,  and  350.  The  distances  determined  be- 
ing, to  those  calculated  upon  the  ground  that  the  orbit  was  a 
circle,  as  1  to  1.0053,  as  1  to  1.0048,  and  as  1  to  1.0021. 

But  Mr.  Vince  says  that  "  Kepler  had  too  good  an  opinion  of 
Tycho's  observations  to  suppose  that  this  difference  might  arise 
from  their  inaccuracy;  and  as  the  distance  between  the  aphelion 
and  perihelion  was  too  great,  upon  the  supposition  that  the  orbit 
was  a  circle,  he  knew  that  the  form  of  the  orbit  must  be  an  oval 
or  ellipse,  with  the  sun  in  one  of  the  foci ;  and  upon  calculating 
the  said  observed  or  determined  distances,  he  found  they  agreed 
together.  He  did  the  same  in  regard  to  other  parts  of  the  orbit,  and 
found  they  all  agreed ;  and  thus  he  pronounced  the  orbit  of  Mars 
to  be  an  ellipse,  having  the  sun  in  .one  of  the  foci. 

Having  determined  this  for  the  orbit  of  Mars,  he  conjectured 
the  same  to  be  true  of  all  the  other  planets,  and  upon  trial  he 
found  it  to  be  so.  Hence  he  concluded  that  the  six  primary 
planets  revolve  about  the  sun  in  ellipses,  having  the  sun  in  one 
of  the  foci." 

Mr.  Vince  says,  "  that  Kepler,  in  calculating  the  eccentricity 


RESPECTING    ELLIPTICAL    ORBITS.  223 

of  the  orbit  of  Mars,  upon  the  hypothesis  that  it  was  a  circle,  and 
on  Tycho's  principle  that  it  had  an  equable  motion  around  a 
point,  (supposing  the  sun  and  such  point  were  bisected  by  the 
centre  of  the  circle,)  and  having  calculated  twelve  oppositions  of 
Mars  observed  by  Tycho,  none  of  which  differed  more  than 
1'  47'' ,  he  found  that  the  hypothesis  did  not  agree  with  the  lati- 
tude observed  in  opposition,  nor  with  the  longitude  out  of  oppo- 
sition, which  differed  sometimes  8'  from  observation;  from 
which  want  of  agreement,  and  from  the  place  of  the  sun,  and 
the  point  around  which  the  motion  was  supposed  to  be  equable, 
not  being  bisected  by  the  centre  when  thus  computed,  he  was 
persuaded  that  the  orbit  of  Mars  was  not  a  circle." 

But  it  is  said  that  the  circle,  which  Kepler  says  so  wrell  repre- 
sented the  twelve  Oppositions  observed  by  Tycho,  and  on  which 
he  based  his  calculations,  had  an  eccentricity  equal  to  18544, 
between  the  sun  and  point  around  which  the  motion  was  suppo- 
sed to  be  equal,  but  he  found  this  quantity  to  be  divided  by  the 
centre  of  the  orbit  as  follows  :  between  the  sun  and  centre  of  the 
orbit,  11332 ;  and  between  the  centre  of  the  orbit  and  sup- 
posed point  of  equable  motion,  7232. 

Now  whether  there  was  error  in  Tycho's  observations, 
whether  there  was  error  or  discrepancy  in  Kepler's  calculations, 
or  whether  they  may  have  been  wrongly  quoted,  I  know  not. 
But  in  one  case  upon  Tycho's  observations.  Kepler  finds  the  ec- 
centricity of  the  orbit  of  Mars  to  be  14140,  (calling  the  distance 
from  our  earth  to  the  sun,  100000 ;)  and  in  another  case  upon  Ty- 
cho's observations,  he  finds  the  distance  between  the  sun  and  cen- 
tre of  Mars'  orbit,  to  be  11332,  —  a  difference  of  near  3000  in  the 
two  calculations,  the  double  of  which  subtracted  from  the  line  of 
the  apsides,  as  by  him  first  determined,  would  serve  doubtless  to 
fully  compensate  his  oval  or  ellipse  into  a  circle,  according  to  his 
determination  of  the  three  distances  in  other  parts  of  the  orbit. 

But  without  more  light  being  thrown  upon  this  intricate  sub- 
ject,—  depending,  as  it  must  of  necessity,  upon  observation 
alone  for  its  support, —  it  is  impossible  to  tell  whether  Kepler,  in 
his  investigations  from  which  emanated  his  elliptical  hypothesis, 
detected  and  rectified  error  to  a  greater  extent  than  he  estab- 
lished it. 

And  notwithstanding  we  have  all  looked  upon  Kepler  as  the 
father  of  physical  astronomy,  and  that  he  has  done  more  in  lay- 
ing its  foundation  than  all  who  have  attempted  its  superstruc- 
ture, and  that  it  is  no  idolatry  to  admire  such  wisdom  in  an 
earthly  shape, —  nevertheless,  as  the  elliptical  hypothesis  seems 
to  be  wholly  abstract  from  any  known  physical  law,  or  necessity 
connected  with  unity  of  purpose  or  harmony  of  design,  depend- 


224 

ing  for  its  support  upon  observation  alone,  let  it  again  be  cor- 
rectly examined,  and  if  found  to  be  true,  it  is  doubtless  of  more 
extended  consequence  than  we  have  heretofore  conceived  it  to  be. 

But  Kepler  himself  was  somewhat  perplexed  with  his  ellipti- 
cal hypothesis;  for,  although  his  first  law,  —  namely,  that  the 
motion  varies  inversely  as  the  distance  varies,  —  furnished  the 
means  for  equating  an  orbit,  yet,  upon  the  elliptical  hypothesis, 
the  finding  of  the  true  anomaly  from  the  mean,  or  the  mean 
anomaly  from  the  true,  was  no  easy  task ;  for  Mr.  Vince  says, 
"that  although  Kepler  suggested  a  problem,  which  still  goes  by 
the  name  of  Kepler's  problem,  for  finding  the  mean  anomaly 
from  the  true,  and  vice  versa,  of  a  planet  revolving  in  an  ellipse, 
still  the  problem  has  remained  unsolved  to  this  day,  otherwise 
than  by  approximation,  notwithstanding  it  has  exercised  the  tal- 
ents of  eminent  mathematicians." 

Such,  then,  is  the  evidence  upon  which  the  elliptical  hypoth- 
esis rests ;  and  if  it  be  of  consequence  to  know  whether  it  be  true 
or  false,  it  is  also  of  consequence  that  observations,  depending,  as 
they  necessarily  must,  upon  much  intricacy  and  exactness,  should 
be  so  critically,  repeatedly  and  carefully  made,  as  shall  leave  no 
room  to  doubt. 

Not  being  able  to  call  to  aid  any  evidence  in  support  of  the 
elliptical  hypothesis,  I  will  endeavor  to  make  some  few  compar- 
isons between  the  ellipse  and  the  eccentric  circle,  as  orbits  in 
which  planets  may  be  conceived  to  revolve,  for  the  purpose  of 
seeing  if  any  light  can  be  shed  upon  the  subject  by  such  a 
course ;  and  here  I  will  remark  that  in  such  examples  or  com- 
parisons, one  and  the  same  central  power  is  conceived  to  be  used 
for  all  the  examples. 

Conceive,  then,  an  eccentric  circular  orbit,  whose  distance  is 
8 ;  of  course,  the  line  of  its  apsides  is  8  ;  and  let  its  perihelion 
distance  be  .25,  and  its  aphelion  distance  be  7.75,  its  motion  at 
perihelion  would  of  course  be  4,  and  its  motion  at  aphelion, 
.2666,  —  or  l-15th  part  as  great  as  at  the  perihelion.  Now  sup- 
pose an  ellipse  to  be  inscribed  in  the  circular  orbit,  their  re- 
spective perihelions  and  aphelions  touching  each  other,  with  the 
sun  in  one  of  the  foci  of  the  ellipse.  That  is,  according  to  the 
idea  of  an  elliptical  orbit,  it  shall  be  as  elliptical  as  it  is  eccen- 
tric. Thus  at  the  perihelion  and  aphelion,  the  rates  of  gravity 
will  be  the  same  in  both  orbits,  but  will  be  greater  in  the  ellipse 
than  in  the  circle  in  all  parts  of  the  orbits  excepting  those  two 
points. 

For  by  Kepler's  third  law,  the  periods  of  the  planets  must  be 
the  same,  because  the  line  of  their  apsides  are  equal ;  at  least, 
according  as  Mr.  Norton  renders  the  law. 


RESPECTING    ELLIPTICAL    ORBITS.  225 

Each  planet  in  respect  to  itself,  would  describe  equal  areas  in 
equal  times ;  nevertheless,  the  planet  revolving  in  the  circle 
would  not  only  pass  over  more  area  than  that  revolving  in  the 
ellipse,  in  the  same  time,  but  more  in  proportion  to  the  velocity 
or  space  passed  over. 

As  the  rate  of  gravity  of  the  planet  revolving  in  the  ellipse 
would  in  all  parts  of  its  orbit,  or  at  all  times,  be  greater  than  that 
of  the  planet  revolving  in  the  circle ;  hence,  its  amount  of  gravity 
in  the  time  of  its  period  would  be  greater  than  that  in  the  circle. 

Hence,  we  have  the  strange  anomaly,  of  two  planets  revolving 
in  equally  eccentric  orbits  around  the  same  central  force,  with 
their  mean  distances  equal ;  their  gravities  equal  at  perihelion 
and  aphelion ;  and  yet  one  of  them  constantly  passing  over  but 
about  two  thirds  of  the  space  of  the  other,  and  a  far  less  pro- 
portion of  area,  in  the  same  time,  and  yet  constantly  urged 
towards  the  central  power  by  a  far  greater  force  of  gravity. 

Now  if  planets  were  made  to  revolve  in  those  two  orbits,  they 
would  have  certain  principles  in  common  with  each  other ;  in 
each  the  product  of  the  mean  distance  by  the  mean  motion, 
divided  by  the  actual  distance  at  any  point  of  the  orbit,  would 
give  the  rate  of  motion  at  such  point;  so  each,  in  respect  to 
itself,  would  describe  equal  areas  in  equal  times,  if  made  to 
revolve  by  a  motion  varying  inversely  as  the  distance;  for  the 
principles  are  only  reciprocal  conditions  of  each  other. 

I  will  now  suppose  two  orbits  situated  similar  to  the  forego- 
ing orbits,  the  lines  of  their  respective  apsides  being  2,  (hence 
the  mean  distance  of  each,  according  to  Newton,  is  1,)  and  let 
their  eccentricities  be  such  that  the  circumference  of  the  ellipse 
is  just  2-3rds  that  of  the  circle ;  for  if  planets  can  revolve  in  all 
kinds  of  ellipses,  however  eccentric,  they  can  revolve  in  such  as 
the  foregoing. 

In  such  case  the  rate  of  motion  in  the  ellipse  in  corresponding 
points  of  the  orbit,  should  always  be  just  2-3rds  as  great  as  that 
in  the  circle,  and  hence  must  be  so  in  their  point  of  mean  mo- 
tion. But  if  it  be  not  so,  we  have  no  criterion  whereby  to  pro- 
portion the  rates  of  motion  between  the  two,  any  more  than  we 
have  whereby  to  obtain  the  true  anomaly  from  the  mean,  or  the 
mean  from  the  true,  of  a  planet  revolving  in  an  ellipse ;  in  fact, 
it  would  be  all  anomaly. 

But  what  seems  somewhat  perplexing  in  respect,  to  the  ellip- 
tical hypothesis,  is,  that  it  seems  always  inclined  to  break  the 
law,  and  will  not  be  subject  to  it ;  but  having  fortified  its  posi- 
tion behind  the  supposed  correctness  of  Tycho's  observations, 
and  Kepler's  calculations  made  from  the  same,  it  has  thus  ac- 
29 


226 

quired  a  kind  of  invulnerableness  to  an  attack,  by  any  legitimate 
kind  of  warfare. 

Thus  by  a  universal  law  of  force  and  motion  based  upon 
Kepler's  great  law,  that  the  squares  of  the  periods  are  as  the 
cubes  of  the  mean  distances,  we  find  that  the  mean  distance  of 
a  planet  divided  by  the  mean  motion  gives  the  period ;  or  the 
mean  distance  divided  by  the  period  gives  the  mean  motion. 
And  although  we  find  this  law  to  prevail  in  a  circular  orbit, 
whether  centric  or  eccentric,  it  is  always  broken  in  the  ellipse, 
whether  the  eccentricity  be  much  or  little. 

Thus,  in  the  two  last-named  orbits,  if  the  periods  and  mean 
distances  must  be  equal,  they  will  each  of  necessity  be  1 ;  and 
if  the  mean  distance  of  each  be  divided  by  the  period,  the  re- 
sult will  be  the  same  in  each,  viz.,  unity  or  1 ;  but  it  will  not 
give  the  mean  motion  in  the  ellipse,  for  that  is  but  2-3rds  as  great 
as  in  the  circle. 

So  if  the  mean  distance  of  each  be  divided  by  the  mean  mo- 
tion, the  period  in  the  circle  will  come  out  but  2-3rds  as  great  as 
that  in  the  ellipse.  So  that  in  either  case,  we  find  error,  and  a 
want  of  conformity  to  the  laws  of  force  and  motion.  But 
suppose  in  the  two  last-named  orbits,  that  both  planets  move 
from  the  perihelion  at  the  same  time,  and  that  at  perihelion,  the 
motion  of  the  planet  in  the  ellipse  is  just  two  thirds  of  that  in 
the  circle. 

Now  while  we  are  making  planets  revolve  as  we  please,  we 
may  imagine  that  the  ratios  of  the  motion  of  the  two,  will  so 
continue  to  aphelion,  and  throughout  their  orbits  ;  and  such 
would  be  the  case  if  both  planets  were  constantly  attracted  by 
the  same  intensity  of  force ;  for  in  such  case,  they  would  pre- 
serve at  the  same  height  the  same  ratios  or  proportions  of  mo- 
tion, whatever  might  be  their  respective  routes  in  arriving  at 
such  height ;  at  least,  according  to  Sir  Isaac  Newton ;  for  Mr. 
Maclaurin  says,  (Book  4th,  Sect.  21,)  "  Our  author  (Sir  Isaac 
Newton)  has  shown  that  if  one  body  move  in  a  curve,  and 
another  ascend  or  descend  in  a  right  line,  acted  on  by  the  same 
gravity,  and  their  velocities  be  equal  in  any  equal  altitudes,  they 
will  be  equal  in  all  other  equal  altitudes."  Hence,  if  it  were 
possible  to  make  the  application  of  what  "  our  author  has  shown," 
(which  by  the  way  it  is  not;  for  I  believe  that  no  one  can  well 
suppose  that  if  the  motions  of  the  two  planets  were  equal  in 
perihelion,  that  they  could  take  their  respective  routes  in  their 
orbits,  having  their  velocities  equal  throughout  their  respective 
orbits,  and  at  the  same  time  perform  their  respective  periods  in 
the  same  time,)  it  would  serve  to  establish  my  premises ;  for  the 
amount  of  gravity  applied  to  the  planet  in  the  ellipse  in  the  time 


RESPECTING    ELLIPTICAL    ORBITS.  227 

of  its  period,  is  vastly  greater  than  what  is  applied  to  the  other ; 
and  hence  I  will  leave  it  to  others,  either  to  reconcile  the  incon- 
gruities, or  to  complete  the  demonstration  by  way  of  reductio 
ad  absurdam. 

The  admiration  which  the  wisdom  of  Kepler  so  justly  merits 
from  the  world,  ought  not  to  be  bestowed  indiscriminately  with- 
out regard  to  truth  and  error ;  and  we  should  as  readily  come 
up  to  the  help,  against  error  which  he  may  have  promulgated, 
as  if  promulgated  by  any  other  person. 

I  have  already  stated  that  his  hypothesis  in  respect  to  ellip- 
tical orbits  is  not  founded  upon  those  immutable  laws,  which 
may  be  mathematically  or  numerically  determined,  as  in  respect 
to  his  other  two  laws,  but  rests  upon  the  supposition  that  Tycho's 
observations,  as  well  as  his  calculations  made  thereon,  were 
wholly  correct;  and  that  there  could  have  been  no  error,  either 
direct  or  collateral  in  either.  Nevertheless,  while  his  other  two 
laws,  which  are  immutable  and  eternal  in  the  heavens,  have  as 
yet  been  more  or  less  misapprehended  and  misapplied,  his 
hypothesis  in  respect  to  an  elliptical  form  of  the  orbits,  has  not 
only  taken  deep  root,  but  has  grown  to  the  overshadowing  of 
every  other  principle  in  respect  to  the  planetary  system. 

Nor  does  it  appear  that  Kepler,  or  others,  who  have  so  indus- 
triously inculcated  the  elliptic  hypothesis,  have  been  aware  that 
it  could  not  be  true,  in  case  the  motion  of  a  planet  in  its  orbit 
varies  inversely  as  the  distance  from  the  sun  varies. 

The  ellipse  or  elongated  circle,  may  vary  from  that  of  the 
circle,  to  one  of  extreme  ellipticity,  in  which  the  two  sides  will 
differ  but  little  from  straight  lines  ;  and  the  distance  from  either 
focus  of  the  ellipse  to  the  nearest  or  adjacent  end  or  vertex,  is 
always  less  than  half  of  the  shortest  axis  or  diameter  of  such 
ellipse ;  and  it  is  gravely  taught  as  a  philosophic  truth,  that  a 
planet  may  revolve  in  an  ellipse,  however  elliptical ;  and  also 
that  the  motion  of  such  planet,  will  vary  inversely  as  the  square 
of  the  distance  from  the  sun  varies. 

Now  it  would  seem  that  the  simple  mention  of  these  two 
positions  in  connection,  would  be  sufficient  to  show  the  import- 
ant error  contained  therein,  without  any  attempt  whatever  at 
further  explanation ;  and  that  Kepler  could  not  have  committed 
the  error,  in  case  the  law  of  falling  bodies  had  then  been  dis- 
closed by  Galileo,  his  contemporary.  But  that  Newton,  who 
seems  not  to  have  been  culpable  for  any  error  or  imposition 
which  he  might  entail  upon  the  learned  world,  should  have 
committed  the  error,  with  the  law  of  falling  bodies  before  him, 
(not  to  say  with  the  full  knowledge  of  the  law  of  falling  bodies,) 
is  to  me  not  at  all  surprising. 


228  ON  KEPLER'S  HYPOTHESIS 

For  notwithstanding  his  attempts  at  a  disclosure  of  the  law 
of  gravity  from  a  consideration  of  the  law  of  falling  bodies, 
(the  very  process  which  should  have  detected  the  fallacy  of  the 
elliptic  orbit,)  the  bungling  and  palpable  errors  therein  com- 
mitted might  well  induce  the  belief  that  no  error  was  too  pal- 
pable or  gross  for  him  either  to  commit  or  overlook. 

The  ellipse,  or  elongated  circle,  may  vary  from  the  circle  by 
any  assignable  quantity ;  for  as  NewTton  would  say,  Neque  novit 
natura  limitum. 

If  a  planet  revolve  in  a  centric  orbit,  (viz.,  with  the  sun  in  the 
centre,)  the  motion  will  be  uniform  ;  the  whole  amount  of  cen- 
tripetal force  during  the  time  of  the  period,  being  expended  in 
producing  the  convergency  of  the  planet ;  and  consequently,  no 
portion  of  the  force  is  expended  in  accelerating  or  retarding  the 
motion  of  the  planet. 

If  such  orbit  shall  become  eccentric,  (still  retaining  its  circu- 
lar form,)  inasmuch  as  the  planet  in  the  course  of  its  revolu- 
tion approaches  and  recedes  from  the  sun,  then  upon  the  prin- 
ciple of  falling  bodies,  its  motion  must  alternately  be  accelerated 
and  retarded,  and  consequently  a  portion  of  the  constant  force 
applied  is  expended  in  such  acceleration  and  retardation,  and 
the  remainder  in  the  convergency. 

Now  the  convergency  of  a  planet  is  no  part  of  the  motion, 
and  may  simply  be  denned  to  be  the  direction  of  the  motion; 
and  so  we  find  the  definition  in  treatises  upon  the  involution  and 
evolution  of  curves,  where  it  is  said  that  "curvature  is  a  change 
of  direction  only,"  although  it  is  frankly  acknowledged  in  those 
treatises,  that  curvature  has  been  illy  defined,  and  the  subject 
very  much  perplexed  in  its  consideration.  Nor  is  it  passing 
strange  that  the  subject  should  have  been  perplexed,  considering 
the  manner  in  which  it  has  been  treated. 

I  have  heretofore  attempted  to  show  that  convergency  is  not 
only  a  proper,  but  an  important  element  in  the  consideration  of 
planetary  motion ;  that  it  is  a  tendency  to  the  same  point  again, 
from  which  the  planet  has  revolved ;  that  in  the  time  of  the 
period,  the  convergency  is  always  360°,  and  consequently  the 
amount  of  convergency  in  the  time  of 'the  period  is  always 
unity,  and  hence  might  be  assumed  as  a  standard  element  in 
any  orbit,  whatever  may  be  the  distance ;  that  it  is  directly  con- 
nected with  planetary  motion  and  the  law  of  gravity ;  and  that 
the  rates  of  motions,  of  gravity,  and  of  convergency,  are  so  pro- 
portioned to  each  other,  and  to  the  period  and  distance,  as  to 
perform  the  grand  results  required  by  the  Keplerian  law.  That 
the  mean  rate  of  convergency  is  always  inversely  as  the  time  of 
the  period,  —  consequently,  inversely  as  the  cube  of  the  square 


RESPECTING    ELLIPTICAL    ORBITS.  229 

root  of  the  distance,  —  it  is  hence,  the  cube  of  the  mean  rate  of 
the  motion,  and  the  cube  of  the  square  root  of  the  mean  rate 
of  the  gravity  or  force,  &c.  And  hence,  the  mean  rate  of 
gravity  or  force,  is  always  a  mean  proportional  between  the 
mean  rate  of  motion,  and  the  mean  rate  of  convergence ;  and 
gravity  throughout  the  period  is  equally  attentive  to  conver- 
gency  as  it  is  to  motion;  and  whether  the  orbit  be  centric 
or  eccentric,  the  element  of  gravity  or  force,  at  all  times  and  in 
all  parts  of  the  orbit,  preserves  the  same  position  in  respect  to 
the  elements  of  motion  and  convergency,  viz.,  a  mean  propor- 
tional between  them ;  for  upon  no  other  condition  can  the 
motion  of  the  planet  vary  inversely  as  the  distance  varies  from 
the  sun ;  nor  should  we  be  much  alarmed  at  the  immutable  pre- 
cision by  which  the  force  is  thus  divided  or  proportioned. 

Suppose,  then,  the  force  not  to  be  thus  proportioned  between 
the  motion  and  convergency ;  that  is,  suppose  the  orbit  to  be 
an  ellipse,  in  which  case  it  is  manifest  that  the  force  cannot  be 
thus  proportioned  to  the  motion  and  convergency.  And  to  lay 
our  premises  broad,  we  will  suppose  such  ellipse  to  possess  very 
great  ellipticity  ;  in  which  case  the  planet,  in  descending  from 
its  aphelion  to  its  perihelion,  shall  describe  a  line  of  but  little 
curvature,  or  varying  but  little  from  a  rectilinear  fall ;  in  which 
case,  the  motion  of  the  planet  at  aphelion  will  be  as  near  to  0  as 
the  ellipticity  of  the  orbit  is  to  extreme  ellipticity. 

Now  the  rates  of  commencement  of  motion,  or  of  continued 
motion  of  two  bodies  falling  from  a  state  of  rest  at  the  same 
time  from  different  distances,  towards  the  same  centre  of  gravity, 
are  inversely  as  the  squares  of  the  distances  at  which  they  com- 
mence their  respective  fall,  for  the  very  reason  that  the  force  of 
gravity  is  inversely  as  the  distance ;  so  also,  the  motion  of  each 
body  from  the  commencement  of  its  fall,  will  vary  inversely  as 
the  square  of  the  distance  varies ;  and  such  would  be  the  condi- 
tion of  a  body  revolving  in  the  extreme  ellipse,  (if,  however, 
such  impossibility  could  be  performed;  but  I  will  not,  like  Mac- 
laurin,  be  wiser  in  respect  to  laws  that  have  no  existence  in 
nature,  than  those  which  have.) 

The  consequence  must  then  necessarily  be,  that  as  we  vary 
from  the  circular  orbit,  whether  centric  or  eccentric,  in  which  the 
motion  varies  inversely  as  the  distance  varies,  and  approximate 
from  the  ellipse  of  least  ellipticity  to  that  of  the  greatest,  we 
must  necessarily  approximate  from  the  inverse  variation  of  the 
motion  to  the  distance,  to  the  inverse  variation  of  the  motion 
to  the  square  of  the  distance. 

I  have,  therefore,  in  my  investigations,  become  rather  satis- 
fied that  no  other  explication  co^Ul  well  be  given  of  the  opera- 


tion  of  force  and  motion  as  applicable  to  a  planet  revolving  in 
an  ellipse,  than  that  given  by  Maclaurin  and  others,  viz.,  that 
the  force  and  motion  alternately  get  the  better  of  each  other. 
And  whether  it  shall  continue  to  be  said,  as  heretofore,  that, 
"  As  it  has  been  proved  that  a  planet  can  revolve  in  an  ellipse, 
having  the  sun  in  one  of  the  foci,  it  consequently  follows  as  a 
general  law,  that  bodies  may  revolve  in  all  kinds  of  ellipses, 
however  eccentric,"  or,  whether  it  shall  hereafter  be  said  that, 
as  it  has  been  shown  that  bodies  cannot  by  the  laws  of  force 
and  motion,  revolve  in  all  kinds  of  ellipses,  however  eccentric, 
it  consequently  follows  that  a  body  cannot,  by  the  laws  of  force 
and  motion,  revolve  in  any  kind  of  ellipse,  —  are  questions 
which  are  submitted  for  consideration. 


CHAPTER    V. 

Outline  of  a  Theory  of  the  Solar  System. 


I  have  already  intimated  that  I  might  offer  an  outline  of  my 
theoretic  notions  in  respect  to  the  great  physical  law  of  the  uni- 
verse, under  its  various  modifications,  in  its  constant  operations 
in  the  adjustment  and  control  of  the  Solar  System;  referring  for 
evidence  in  support  of  such  theory,  so  far  as  I  am  able  to  avail 
myself  of  them,  to  such  facts,  phenomena  and  analogies  in  na- 
ture, already  before  the  world,  as  will,  in  my  opinion,  best  serve 
to  sustain  or  support  such  theory  or  hypothesis. 

In  assigning  causes  to  the  phenomena  of  nature,  perhaps  we 
cannot  well  adopt  a  better  criterion  by  which  to  be  governed, 
than  that  contained  in  the  first  two  rules  laid  down  by  Sir 
Isaac  Newton,  namely :  "  No  more  causes  are  to  be  admitted 
than  are  sufficient  to  explain  the  phenomena;"  and  2d,  "Of 
effects  of  the  same  kind,  the  same  causes  are  to  be  assigned,  as 
far  as  it  can  be  done."  And  it  might  be  well  for  the  advance- 
ment of  science  and  philosophy,  were  these  rules  more  strictly 
adhered  to.  But  we  are  all  quite  too  apt  to  deviate  from  those 
rules  which  we  may  have  prescribed  for  our  future  conduct. 

But  in  attempting  to  theorize  where  the  immutability  of  num- 
bers does  not  come  directly  to  our  aid,  there  are  many  things  as 
yet  too  high  for  our  understanding,  —  too  vast  for  our  compre- 
hension,—  the  causes  too  remote  to  be  assigned,  and  the  opera- 
tions too  abstruse  and  evasive  in  their  detection,  to  be  well  de- 
fined or  explained,  without  more  facts  and  data  than  the  accu- 
mulated knowledge  of  the  world  is  yet  in  possession  of,  to  enable 
us  to  form  a  theory  (even  of  the  Solar  System,)  that  is  not  in 
many  respects  liable  to  constant  demolition,  arising  from  the  dis- 
covery of  new  facts  and  analogies  ;  all  of  which,  doubtless,  tends 
to  show  the  simplicity,  harmony,  reciprocity  and  analogy  which 
pervade  all  the  laws  of  nature,  and  their  operations. 


232  OUTLINE    OF    A    THEORY 

I  will  not  here  make  the  sacrilegious  attempt  to  theorize  upon 
the  primitive  formation  of  the  universe,  as  many  have  done, 
for  the  purpose  of  accounting  for  all  things;  or,  at  least,  of 
being  understood  by  others  to  have  done  so, —  as  how  adventitious 
nature,  with  her  gravity  on  hand,  accumulated  her  atomic  matter 
into  its  present  aggregate  masses,  from  which  gravity  is  con- 
stantly emanating  in  search  of  prey,  and  that  nothing  but  the 
speed  of  the  smaller  bodies  prevents  their  being  dragged  to  the 
larger,  —  or  the  no  less  vague,  unwarranted  and  uncalled  for  as- 
sumption that  all  the  revolving  bodies  of  the  solar  system  were 
originally  projected  in  tangents  to  their  respective  orbits,  with 
such  forces  as  would  produce  the  present  momenta  of  the  revolv- 
ing body  in  that  point  of  the  orbit  where  it  was  first  taken  charge 
of  by  gravitation,  whatever  point  in  the  orbit  that  might  have  been 
—  so  that  all  the  people  might  know  by  what  means  the  planets 
and  other  revolving  bodies  first  got  under  way.  But  as  none  of 
these  projections  have  happened  in  our  day,  our  faith  in  the  hy- 
pothesis (if  we  have  any)  must  rest  upon  a  supposition  or  ap- 
prehension which  is  of  no  use  whatever  to  science,  as  it  is  not  a 
phenomenon  of  nature  or  of  nature's  laws;  we  see  it  not,  we 
know  it  not,  and  at  best  it  is  but  an  attempt  to  show  a  necessity, 
consonant  with  human  contrivance,  which  Deity  was  under,  of 
making  use  of  certain  means  for  the  accomplishment  of  his  pur- 
poses in  creating  and  upholding  the  universe. 

Not  so  with  the  laws  of  nature,  which  are  continually  operat- 
ing, and  the  phenomena  which  they  produce,  and  which  it  is 
our  highest  privilege  to  investigate,  thereby  exalting  the  human 
mind  by  a  more  comprehensive  view  or  understanding  of  cause 
and  effect,  which,  more  than  anything,  distinguishes  man  from 
the  brutes. 

Nor  will  I  undertake  to  fancy  that  this  earth,  at  some  former 
period,  was  detached  from  the  sun  in  one  molten  mass,  the  out- 
side of  which  as  yet  is  but  just  cooled  to  a  crust,  and  that  we 
are  yet  in  danger  of  having  it  broken  up  in  some  terrible  erup- 
tion, and  we  plunged  into  the  eternal  fires,  &c. 

But  I  would  rather  confine  my  speculations  to  the  solar  sys- 
tem, as  it  now  exists,  and  as  it  now  appears  to  be  governed  by 
those  immutable  laws  founded  in  the  fitness  of  things,  and  from 
which  all  the  phenomena  of  nature  are  doubtless  results,  effects 
and  consequences. 

Of  gravitation,  or  the  attraction  of  gravitation,  we  at  present 
know  but  little,  otherwise  than  through  its  effects  and  operations 
upon  palpable  matter,  or  matter  in  aggregate  form.  Nor  is  our 
philosophy  sufficiently  astute  to  reason  intelligibly  upon  the  ope- 
rations of  gravity,  or  attraction,  upon  two  inconceivably  small 

'  •  •  • : 


OF    THE    SOLAR    SYSTEM.  233 

corpuscles  or  physical  atoms,  and  thence  to  transfer  the  analogies 
of  those  operations  to  that  of  tangible  masses  of  matter,  as  has 
been  attempted  and  pretended  by  high  philosophers. 

Nevertheless,  we  have  sufficient  facts,  data  and  phenomena, 
resulting  from  the  laws  of  gravitation  or  attraction,  from  which 
to  deduce  many  important  results,  if  analogically  considered, 
amongst  which  is  that  of  the  ratio  in  respect  to  distance,  by  which 
the  attraction  of  gravitation  operates  upon  matter.  But  in  what 
way  or  manner  the  influence  of  gravitation  is  exerted  on  matter, 
has  perhaps  never  been  well  defined  or  determined.  Neverthe- 
less, it  is  not  absolutely  necessary  that  we  should  conceive  it  to 
operate  through  absolute  space,  without  a  material  medium  in 
and  through  which  to  operate ;  and  which  medium,  however  at- 
tenuated, has  a  density  varying  in  some  given  ratio  to  the  dis- 
tance from  bodies  of  aggregate  matter,  namely,  in  the  same  ra- 
tio as  that  of  the  force  of  gravitation. 

Perhaps  there  are  but  few  astronomical  facts  better  ascertained 
from  observation  and  calculation,  than  that  the  attractive  powers 
exerted  by  the  several  primary  planets  of  the  solar  system,  are 
as  their  bulks  or  magnitudes  directly,  and  their  inverse  distances 
from  the  sun  conjointly.  And  if  such  be  the  case,  it  furnishes 
strong  presumption,  at  least,  that  their  attractive  powers  are  not 
innate,  and  that  attraction  or  gravitation  is  not  an  essential  prop- 
erty of  matter ;  but  that  the  planets  of  the  system  obtain  their 
gravitating  and  attracting  powers  from  the  circumstance  of  their 
being  within  the  attracting  power  of  the  sun,  in  the  same  man- 
ner as  soft  iron  obeys  the  magnetic  influences  when  brought 
near  a  permanent  magnet  or  an  electro-magnetic  battery. 

I  know  that  Sir  Isaac  Newton,  having  adopted  the  theory  of 
the  equal  and  innate  gravity  of  all  matter,  and  observing  that  the 
attracting  power  of  Jupiter  and  Saturn  upon  their  respective  sat- 
ellites was  not  as  great  in  proportion  to  their  apparent  magni- 
tudes as  that  of  the  earth  upon  the  moon,  adopted  the  concomi- 
tant hypothesis,  that  those  primary  planets  are  less  dense  than 
the  earth  is.  And  such  has  since  been  the  progress  in  ascertain- 
ing the  different  densities  of  the  primary  planets,  that  Euler  cal- 
culated their  densities  to  be  inversely  as  their  distances  from  the 
sun  ;  and  the  astronomer  and  mathematician,  LaGrange,  long 
since  adopted  the  same  ratio,  as  a  standard  from  which  to  de- 
termine their  disturbing  forces  upon  other  matter  ;  and  after 
much  observation  on  the  subject,  remarked  that  he  saw  no  cause 
for  altering  that  ratio,  believing  it  to  be  true. 

To  be  sure,  the  present  Herschel  supposed  the  planet  Hers- 
chel  to  have  a  greater  density  than  would  be  given  it  by  su^h 
oO 


234  OUTLINE    OF    A    THEORY 

ratio,  but  modern  observations  have  determined  that  he  overrated 
the  density  of  Herschel. 

But  I  know  of  no  analogies  in  nature  to  sustain  the  hypothe- 
sis of  the  different  densities  of  the  planets,  and  shall  therefore  en- 
deavor to  seek  for  the  cause  of  the  phenomena  upon  which  such 
hypothesis  is  based,  in  the  analogies  arising  from  observed  phe- 
nomena, as  well  as  such  direct  proof  as  may  have  been  adduced 
by  those  who  have  been  most  forward  in  establishing  and  sus- 
taining the  hypothesis  of  the  different  densities  of  the  planets. 
But  surely  there  can  be  no  analogy  between  the  hypothesis  that 
the  force  of  gravity  varies  inversely  as  the  square  of  the  distance, 
and  that  which  assumes  the  densities  of  the  planets  to  be  in- 
versely as  their  distances  from  the  sun.  But  that  there  are  anal- 
ogies between  the  hypothesis  that  the  force  of  gravity  varies  in- 
versely as  the  distance,  and  that  the  planets  vary  in  their  re- 
spective gravities  in  proportion  to  their  respective  magnitudes, 
in  the  inverse  ratio  to  their  respective  distances  from  the  sun,  I 
will  endeavor  to  show. 

In  respect  to  the  Newtonian  law  of  gravity,  namely,  that  grav- 
ity varies  inversely  as  the  square  of  the  distance,  we  find  in 
Vince's  History  of  Astronomy,  Vol.  2,  p.  279,  the  following, 
namely : 

"  That  in  1684,  Dr.  Halley  turned  his  thoughts  to  the  relation 
between  the  periodic  times  and  distances  of  the  planets,  and 
concluded  from  it  that  the  centripetal  force  must  vary  inversely 
as  the  squares  of  the  distances  ;  but  not  being  able  to  prove  it, 
he  applied  to  Mr.  Hook  and  Sir  Christopher  Wren  ;  they,  how- 
ever, not  being  able  to  give  him  satisfaction,  he  went  to  Cam- 
bridge, to  Mr.  (afterwards  Sir  Isaac)  Newton,  who  soon  gave 
him  a  proof  of  his  position." 

And  this  a  proof ]  as  also  that  of  Newton's  theory  of  universal 
gravity,  seems  not  only  to  have  satisfied  the  mind  of  Dr.  Halley, 
who  was  searching  for  truth,  and  longing  to  be  satisfied,  but 
also  that  of  the  world  in  general ;  and  it  appears  that  no  farther 
inquiry  was  made  upon  the  subject.  Thus  we  find  in  the  Amer- 
ican Encyclopaedia,  under  the  head  of  Gravity,  that  the  author 
asserts,  that  gravity  appears  to  be  an  essential  property  of  mat- 
ter;  that  it  is  proportional  to  the  masses  of  bodies;  that  it 
varies  inversely  as  the  square  of  the  distance  in  proceeding 
from  the  surface  of  the  body  outwards,  or  from  its  centre ;  that 
it  varies  directly  as  the  distance,  in  descending  from  the  surface 
to  the  centre  in  uniform  spherical  bodies ;  and  that  it  is  trans- 
mitted instantaneously  from  one  body  to  another,  &c.  Some  of 
which  allegations  are  certainly  worthy  of  consideration  and 
remark  in  due  time. 


OF    THE    SOLAR    SYSTEM.  235 

But  a  vast  amount  of  what  is  generally  conceived  by  the 
world  to  be  evidence  of  the  law  of  gravity,  as  declared  or  pro- 
mulgated by  Sir  Isaac  Newton,  may  be  found  in  Vince's  Physi- 
cal Astronomy,  Vol.  2,  Chap.  31,  under  the  head  of  "  The  gen- 
eral principles  of  centripetal  forces,"  in  which  frequent  refer- 
ence is  made  to  the  Principia,  as  authority ;  and  in  which,  to 
me,  the  reasoning  appears  to  be  vastly  more  deep  or  dark,  than 
profound ;  as  by  attempting  to  geometrize  upon  a  geometric 
point,  or  upon  the  law  of  gravity  which  two  geometric  points 
or  physical  atoms  will  regard  in  consequence  of  their  innate 
gravity  upon  each  other,  aided  by  the  use  of  expressions  like 
the  following:  the  limiting  state,  the  limiting  ratio,  a  ratio  of 
equality,  ultimately  proportional,  &c.,  —  which  being  applied  to 
the  circular  orbit,  the  transition  is  then  made  to  the  ellipse,  the  par- 
abola, and  the  hyperbola.  (And,  perhaps,  no  one  will  doubt  but 
that  the  same  law  of  gravity  which  operates  in  the  circle,  will 
operate  also  in  the  ellipse,  the  parabola,  and  the  hyperbola,  so 
far,  at  least,  as  distance  is  concerned.)  And  in  the  result,  we  find 
Mr.  Vince  coming  to  the  following  conclusion,  viz. :  "  Hence, 
we  conclude  that  each  planet,  is  attracted  to  the  sun  by  a  force 
which  varies  inversely  as  the  square  of  the  distance  of  its 
centre,  and  that  the  constituent  particles  also  attract  each  other 
by  a  force  varying  according  to  the  same  law,"  and  that  the 
measure  of  force  will  be  obtained  by  dividing  the  square  of  the 
motion  by  the  distance ;  which,  if  true,  would  consequently  make 
the  force  inversely  as  the  square  of  the  distance ;  which  explan- 
ation has  too  well  satisfied  the  world. 

I  will  now  proceed  to  a  consideration  of  the  attractive  power 
of  the  planets  as  proportioned  to  their  magnitudes  and  distances 
from  the  sun. 

Mr.  Vince  says,  "  That  to  understand  the  principle  upon 
which  this  determination  rests,  we  may  observe  that  the  effect 
of  attraction  at  equal  distances  will  be  in  proportion  to  the 
quantity  of  matter  in  the  attracting  body ;  and  at  different  dis- 
tances, as  the  quantity  of  matter  and  the  inverse  square  of  the 
distance  conjointly.  The  quantity  of  matter  is  also  in  propor- 
tion to  the  magnitude  of  the  body  and  its  density  conjointly.  If, 
therefore,  we  know  the  effects  of  the  attraction  of  different 
bodies,  together  with  their  magnitudes,  we  can  find  their  densi- 
ties, and  thence  their  quantities  of  matter." 

And  this  is  true,  in  case  the  theory  of  equal  innate  gravity  of 
matter  be  true  ;  in  which  case,  the  density  of  a  central  body,  as 
of  Jupiter  for  instance,  will  be  numerically  expressed  in  the 
reciprocal  of  the  product  of  the  magnitude  of  the  central  body 
by  the  square  of  the  period  of  the  revolving  body ;  which  meth- 


236  OUTLINE    OF    A    THEORY 

od,  if  the  principle  were  but  correct,  would  furnish  a  very  easy 
method  of  obtaining  the  different  densities,  (so  called,)  or  the 
proportional  quantities  of  matter  of  those  planets  having  satel- 
lites revolving  around  them. 

But  the  world  are  wishing  for  the  best  evidence  by  which  to 
account  for  the  phenomenon,  that  the  power  of  a  planet  to 
attract  its  satellite  varies  in  proportion  to  the  magnitude  of  the 
planet,  in  the  inverse  ratio  to  its  distance  from  the  sun.  And 
for  the  purpose  of  presenting  some  natural  and  consequential 
evidence  upon  the  subject,  I  will  present,  in  short,  a  method 
which  I  would  pursue  in  the  deduction  of  consequences  from  ob- 
served phenomena,  with  a  view  to  some  unity  of  purpose ;  and 
to  this  end,  I  would  assume  the  distance  of  the  moon  from  the 
earth  to  be  unity  or  1.  In  which  case,  if  the  distance  of  the 
earth  from  the  sun  be  95,000,000  miles,  and  the  distance  of  the 
moon  from  the  earth  be  237,500  miles  from  the  earth,  (as  esti- 
mated by  some,)  the  proportional  distances  between  the  earth 
and  moon,  and  between  the  earth  and  sun,  will  be  as  1  to  400, 
which  I  will  assume  for  the  sake  of  round  numbers.  And  as 
the  relative  motion  of  the  moon  in  its  orbit  about  the  earth,  is 
to  that  of  the  motion  of  the  earth  in  its  orbit  about  the  sun,  as 
1  to  29.9207,  the  square  of  which  is  895.2428,  this  multiplied 
by  400,  (the  relative  distance  of  the  sun,)  gives  358099,  and  a 
fraction,  as  the  relative  power  of  the  sun  to  that  of  the  earth;  — 
that  of  the  earth  being  called  unity.  But  the  exact  proportion 
between  the  power  of  the  earth  and  sun  must  depend  upon 
getting  the  true  relative  distances  between  the  earth  and  moon, 
and  between  the  earth  and  sun,  which,  perhaps,  cannot  be  ob- 
tained to  a  great  degree  of  exactness. 

So,  if  we  take  the  mean  between  the  estimates  of  Cassini  and 
Sir  Isaac  Newton,  as  the  mean  distance  of  Jupiter's  fourth  satel- 
lite from  Jupiter,  its  relative  distance  from  Jupiter  to  that  of 
Jupiter  from  the  sun,  will  be  as  1  to  427,  and  the  relative  mo- 
tion of  such  satellite  in  its  orbit  to  that  of  Jupiter  in  its  orbit, 
will  be  as  1  to  1.6463,  — in  which  case  the  relative  power  of  at- 
traction of  Jupiter  to  that  of  the  sun  would  be  as  1  to  1157.8501. 
Hence,  the  relative  power  of  attraction  of  the  earth  to  that  of 
Jupiter  (calling  the  distance  of  the  moon  from  the  earth  237,500 
miles ;  or  rather,  calling  the  relative  distance  of  the  moon  from 
the  earth  to  that  of  the  earth  from  the  sun  as  1  to  400)  would 
come  out  as  1  to  a  little  more  than  309.  Or,  if  we  call  the 
relative  distance  of  the  moon  from  the  earth  to  that  of  the  earth 
to  the  sun,  as  1  to  391,  then  upon  the  foregoing  calculation,  the 
relative  force  to  that  of  Jupiter  would  be  about  as  1  to  312, 
which  is  in  general  about  the  proportion  assumed.  But  it  must 


OF    THE    SOLAR    SYSTEM.  237 

be  manifest  that  the  estimates  of  relative  power  must  depend 
upon  the  relative  distances  we  assume ;  and  those  cannot  be 
ascertained  to  a  very  great  degree  of  exactness. 

The  foregoing  method  goes  upon  the  principle  that  the  force 
exerted  by  the  sun  upon  the  planets  is  in  the  inverse  ratio  to 
their  respective  distances  from  the  sun;  and  hence  that  the  mean 
force  exerted  by  the  sun  upon  a  planet  is  the  square  of  the  mean 
motion  of  the  planet;  and  that  such  is  the  case  in  respect  to  the 
earth  and  moon,  as  in  respect  to  Jupiter  and  its  satellites.  And 
hence,  that  the  force  exerted  by  the  sun  upon  the  earth  at  the 
distance  400  is  a  little  over  895  times  as  great  as  that  exerted  by 
the  earth  upon  the  moon  at  the  distance  1. 

Thus  the  force  exerted  is  always  inversely  as  the  distance  ; 
and  hence  the  proportional  power  of  gravity  or  attraction  of  the 
respective  planets  of  the  solar  system,  are  as  their  respective 
magnitudes  and  their  inverse  distances  conjointly.  That  is,  if 
one  planet  be  twice  the  distance  of  another  planet  from  the  sun, 
its  power  of  attraction  will  be  but  just  one  half  as  great  in  pro- 
portion to  its  magnitude  as  that  of  the  other. 

And  when  inquiry  into  the  phenomena  of  nature  presents 
and  submits  itself  directly  to  mathematical  investigation  and 
solution,  corroborated  by  well  known  laws,  —  taking  for  data 
such  elements  only  as  would  most  naturally  assign  the  true 
cause  from  the  result  produced,  without  seeking  further  for  an 
hypothesis  by  which  to  account  for  the  cause  of  the  phenomena; 
we  may  feel  much  confidence  in  a  conclusion  which  so  naturally 
follows  from  such  investigation. 

Thus,  the  square  of  the  mean  motion  of  a  revolving  body 
multiplied  into  its  mean  distance  from  the  central  body,  gives  the 
ratio  of  force  between  the  two  bodies.  Or  thus  :  if  the  relative 
motion  of  a  planet  compared  with  the  relative  motion  of  its 
satellite,  be  multiplied  into  the  relative  distance  of  the  planet 
from  the  sun,  compared  with  the  distance  of  the  satellite  from 
the  planet,  we  thereby  obtain  the  relative  power  of  the  sun  com- 
pared with  that  of  the  planet. 

But  as  the  mean  motion  of  a  planet  is  inversely  as  the  square 
root  of  the  mean  distance  from  the  sun,  how  the  fact  should 
have  escaped  observation,  that  the  mean  force  is  the  square  of 
the  mean  motion,  and  consequently  inversely  as  the  distance, 
I  know  not;  for  the  world  has  certainly  acknowledged  those 
principles  in  the  methods  they  have  pursued  for  finding  what 
they  have  conceived  to  be  the  different  densities  of  the  planets. 

Thus,  they  make  the  mean  motion  of  a  satellite,  the  square 
root  of  the  mean  force  of  the  planet,  exerted  upon  the  satellite  ; 
and  so  do  I. 

So  if  there  were  three  planets,  A,  B,  and  D,  of  equal  magni- 


238  OUTLINE    OP    A    THEORY 

tudes,  revolving  about  the  sun,  A  at  the  distance  1,  B  2,  and 
D  4  ;  each  having  a  satellite  revolving  around  it  at  like  distances, 
one  with  the  other  ;  in  such  case,  by  my  calculation,  as  well  as 
by  that  made  by  the  world  in  their  investigations,  the  attractive 
powers  of  the  planets  compared  with  each  other,  will  be  inverse- 
ly as  the  squares  of  the  periods  of  their  respective  satellites  ;  and 
upon  this  ground  the  world  have  placed  the  comparative,  or 
rather  relative  attracting  powers  of  the  earth  and  Jupiter ;  and 
so  have  I. 

The  consequence  then  is,  that  the  reciprocal  of  the  square  of 
the  distance  of  a  satellite,  divided  by  the  square  of  the  period  of 
the  satellite,  gives  the  attractive  power  of  the  primary. 

Then  these  being  facts  and  results  from  all  calculation,  it  only 
remains  that  the  true  cause  should  be  assigned  to  the  phenom- 
ena thus  observed. 

But  whatever  may  be  the  cause  why  the  planets  exert  a  force, 
thus  proportioned  to  their  respective  distances  from  the  sun, 
upon  their  respective  satellites,  we  do  find  that  the  force  thus 
exerted,  is  the  square  of  the  motion  produced  in  the  satellite ; 
thus  emphatically  determining  line  law  of  force  to  be  inversely  as 
the  distance  ;  and  this  goes  far  towards  overturning  that  fortuitous 
notion,  that  matter,  gross  matter,  contains,  or  is  possessed  within 
and  of  itself,  all  the  requisite  powers  and  principles  for  its  gov- 
ernment and  control  throughout  eternity. 

Thus  do  we  find  that  not  only  is  the  force  of  gravity  inversely 
as  the  distance ;  but  that  a  primary  planet  only  exerts  a  force 
upon  its  satellite  in  the  inverse  ratio  to  the  distance  of  such 
primary  from  the  sun ;  which  fact  I  shall  make  use  of  as  an 
unequivocal  inference  hereafter. 

I  have  already  intimated,  that  from  the  many  analogies  which 
magnetism  bears  to  that  of  attraction  in  the  solar  system,  it  de- 
serves consideration  ;  especially  in  the  analogy  presented  in  the 
powers  of  attraction  of  the  planets  as  proportioned  to  their 
respective  bulks  and  distances  from  the  sun ;  and  from  which 
analogies,  much  evidence  is  certainly  derived  to  justify  the 
hypothesis,  that  the  sun  is  the  great  magnetizer  of  the  bodies 
which  compose  the  solar  system  ;  and  that  the  primary  planets 
possess  no  other  attracting  power  than  that  communicated  to 
them  by  the  sun;  and  also,  that  the  attracting  power  of  the  pri- 
mary planets  is  in  the  inverse  ratio  to  their  respective  distances 
from  the  sun,  and  their  respective  bulks  or  magnitudes  conjoint- 
ly ;  and  that  the  density  and  intensity  of  what  I  will  here  call 
the  magnetic  atmosphere  of  the  sun,  varies  inversely  as  the 
distance  varies  from  the  sun. 

It  could  scarcely  be  expected  from  our  present  limited 
knowledge  of  magnetism,  (even  since  the  discoveries  of  Galvani, 


OF    THE    SOLAR   SYSTEM.  239 

and  Oerstead,)  that  we  should  be  able  to  trace  out  all  the  analo- 
gies existing  in  its  various  forms  and  modifications;  —  or  of 
solar  magnetism,  in  all  its  various  forms  and  modifications ;  but 
nevertheless,  there  would  seem  to  be  enough  under  all  circum- 
stances, to  make  it  more  than  probable,  that  the  laws  of  magnet- 
ism are  precisely  those  of  solar  attraction. 

Although  we  may  well  suppose  the  sun  to  be  a  permanent 
magnet,  yet  perhaps  there  are  no  direct  means  for  determining 
whether  the  magnetism  of  the  planets  is  permanent,  or  whether 
they  are  only  dependent  upon  the  sun  for  their  magnetism, 
while  lying  within  his  magnetic  influence,  in  the  same  manner 
as  are  particles  of  iron  filings,  which  become  magnetic  and  attract 
each  other  when  brought  within  the  influence  of  a  permanent 
magnet;  or  like  soft  iron  when  under  the  influence  of  the  electric 
or  galvanic  fluid ;  in  which  cases  the  magnetism  ceases  when  the 
inducing  fluid  is  withdrawn. 

If  then  the  attractive  power  of  a  planet  is  not  an  innate 
principle  of  matter,  but  is  only  induced  by  the  magnetic  influ- 
ence of  the  sun  or  centre  of  attraction,  in  the  manner  in  which 
iron  filings  are  operated  upon  when  under  the  magnetic  influ- 
ence, or  soft  iron,  when  operated  upon  by  the  Galvanic,  or 
Electro-magnetic  fluid ;  then  a  planet  revolving  in  an  eccentric 
orbit,  would  possess  a  greater  power  of  attraction  when  in 
perihelion,  than  when  in  aphelion  ;  and  its  increasing  force  and 
motion  while  passing  from  its  aphelion  to  its  perihelion,  would 
tend  to  dilate  the  orbit  of  its  satellite ;  and  the  reverse  while 
proceeding  from  perihelion  to  aphelion ;  and  were  the  orbit  of 
such  planet  sufficiently  eccentric  for  determining  so  nice  a  test, 
doubtless,  the  oscillations  of  a  pendulum  of  a  given  length,  or  of 
a  compass-needle,  would  be  more  rapid  when  the  planet  was  in 
perihelion,  than  when  in  aphelion. 

So  likewise,  if  a  soft  piece  of  iron  be  made  to  revolve  about 
a  strong  magnet,  and  within  its  magnetic  influence,  its  power  of 
attraction  will  vary  in  some  inverse  ratio  to  its  distance ;  but 
I  do  not  know  but  that  it  would  be  as  dense  in  one  part  of 
its  orbit  as  in  another. 

I  think  then  there  are  analogies  and  unequivocal  inferences  in 
support  of  the  hypothesis  that  attraction  is  not  necessarily  an 
innate  or  inherent  principle  of  matter;  but  I  know  of  no  anal- 
ogies to  support  the  Newtonian  hypothesis  of  universal  gravity 
or  attraction. 

But  perhaps  those  analogies  between  magnetism  and  solar 
attraction,  which  may  serve  to  explain,  (at  least,  theoretically,) 
the  more  hidden  or  occult  recesses  of  nature,  depend  upon 
the  polarity  of  magnetism  in  all  its  various  forms  and  modi- 
fications ;  and  here  we  bring  directly  to  our  aid,  the  phenomena 


240  OUTLINE    OF    A    THEORY 

exhibited  by  electricity  and  Galvanism,  which  appear  to  be  so 
intimately  connected  with  magnetism  as  to  induce  the  belief 
that  they  are  but  modifications  of  one  and  the  same  great  law, 
but  presenting  somewhat  different  phenomena  under  different 
circumstances  and  modifications  ;  and  that  the  magnetic  fluid  of 
the  solar  system  is  the  medium  in  and  through  .which  all  their 
various  operations  take  place,  either  in  the  government  of  inert 
matter,  or  in  the  production  of  organic  matter,  and  in  the  con- 
trol of  final  causes. 

Each  of  these  modifications,  (if  such  they  be,)  has  its  poles, 
or  inverse  influences,  tending,  nevertheless,  to  a  junction  with 
an  ardency  or  intensity  corresponding  with  the  degree  to  which 
they  appear  to  have  been  separated ;  which  poles,  or  adverse 
influences,  are  so  inverse  in  their  operations,  as  often  to  perplex 
the  mind  in  respect  to  them.  Thus  it  is  said,  "  what  we  call  the 
north  pole,  (of  the  magnet,)  because  attracted  by  the  north  pole 
of  the  earth,  is,  for  this  very  reason,  the  south  pole  of  the  mag- 
net ; "  going  upon  the  ground  or  well  known  fact,  that  poles  of 
the  same  kind  repel  each  other,  while  those  of  opposite  kinds 
attract  each  other  ;  and  why  this  circumstance  has  not  given  rise 
to  a  different  method  of  investigating  the  operations  of  terres- 
trial magnetism,  I  know  not. 

I  know  it  has  generally  been  supposed  that  the  earth  was  a 
great  mass  of  inert  matter;  but  at  the  same  time  possessed 
of  power  over  other  inert  matter,  by  means  of  an  innate  or 
inherent  principle  of  attraction  ;  and  that  from  some  unaccount- 
able circumstance  attached  to  this  mass  of  dead  matter,  it  has 
two  poles,  a  north  and  a  south,  with  a  magnetic  equator,  &c. 
Some  who  have  believed  in  the  volcanic  theory  of  internal  fires, 
as  connected  with  the  Huttonian  theory  of  geology,  have  been 
apprehensive  lest,  those  fires  were  not  yet  sufficiently  exhausted,  to 
prevent  danger  from  a  breaking  through  of  the  -crust  in  some 
volcanic  eruption,  or  violent  earthquake.  And  many  a  curious 
calculation  has  been  gone  into,  to  show  how  long  it  would  take 
a  mass  of  matter  as  large  as  the  earth,  to  finally  cool  through 
and  become  one  solid  mass. 

Some  have  imagined  that  there  was  latent  heat  or  caloric 
enough  combined  with  the  solid  matter  of  the  earth,  to  reduce 
the  whole  mass  to  what  they  conceive  to  be  its  primitive  gasses, 
(should  that  heat  once  become  active,)  and  again  fill  a  wondrous 
space  with  its  rare  and  primitive  elements ;  while  others  have 
been  perplexed  with  the  geology  of  the  earth ;  the  inequalities 
of  its  surface ;  the  vast  apparent  changes  that  have  been  going 
on  in  time,  at  and  near  the  surface ;  the  vast  regions  that  have 
manifestly  been  raised  from  under  the  waters  of  the  ocean,  even 
into  elevated  mountains,  &c, ;  many  of  which  phenomena  are 


OF    THE    SOLAR   SYSTEM.  241 

truly  wonderful  when  unaccounted  for ;  and  that  they  are  satis- 
factorily accounted  for  by  the  causes  which  have  generally  been 
assigned,  will  not  be  contended  for  by  a  large  portion  of  the 
community. 

But  had  geologists,  —  while  endeavoring  to  assign  causes  for 
those  phenomena,  in  lieu  of  placing  the  adverse  poles  of  the 
magnetic  fluid,  (in  its  various  modifications,)  upon  the  surface  of 
the  earth,  where  they  appear  to  have  but  little  efficiency  in  the 
great  operations  of  nature,  or  but  little  use.  except  in  directing 
the  compass  needle,  —  only  placed  them  in  contact,  at  the  centre 
of  the  earth,  they  would  at  once  have  assigned  a  cause  sufficient 
to  account  for  all  the  phenomena  that  have  so  much  perplexed 
them,  by  establishing  a  laboratory  sufficient  for  all  the  internal 
operations  of  the  earth ;  as  well  as  to  account  for  most  of  the 
geological  phenomena  upon  and  near  its  surface.  The  intensity 
of  heat  would  then  pervade  the  whole  internal  region  of  the  earth, 
and  fuse  the  whole  internal  mass.  There  would  be  the  process 
of  combining,  separating,  and  mineralizing  the  various  metals, 
minerals,  and  other  substances,  to  the  utmost  capacity. 

As  certain  portions  of  the  solid  parts  of  the  earth  became  thin- 
ner, by  the  fusion  of  its  inner  parts,  they  would  yield  to  internal 
pressure  and  gradually  recede  from  the  centre ;  while  the  thicker, 
and  less  fused  portions,  would  gradually  become  central  por- 
tions; until  their  inner  portions  in  turn,  should  become  molten, 
while  other  portions  were  cooling  and  becoming  thicker;  and 
by  thus  alternating,  (but  without  much  apparent  uniformity,) 
would,  in  connection  with  the  volcanic  flues  or  vents  to  the  in- 
ternal fires,  which  occasionally  and  alternately  pervade  the  sur- 
face of  the  earth  for  periods  of  time,  cause  all  the  phenomena 
upon  the  surface;  as  that  of  the  inequalities  upon  its  surface, 
geological  strata,  the  veins  of  metals,  &c.  This  theory  would 
most  admirably  account  for  the  independent  coal  formations 
which  are  found  to  exist  at  and  near  the  surface  of  the  earth, 
more  commonly  in  latitudes  remote  from  the  equator,  and 
which  are  doubtless  caused  by  deposits  of  vegetable  matter, 
conveyed  from  towards  the  equator  by  ocean  currents,  as  that  of 
the  Gulf-stream,  for  instance,  which  in  process  of  time,  under 
pressure  and  heat,  is  converted  into  coal,  and  thence  raised 
above  the  ocean  for  the  future  use  of  man. 

This  gradual  rising  and  falling  of  different  portions  of  the 
crust,  (and  with  such  uniformity  only  as  all  the  circumstances 
would  permit,  some  perhaps  being  raised  or  sunk  several  times, 
while  others  were  stationary,  or  nearly  so,  together  with  the 
alteration  of  the  internal  portions  of  the  solid  parts,  by  the  action 
of  heat,  as  well  as  by  new  volcanic  vents  produced,  or  old  ones 
31 


242  OUTLINE    OF    A    THEORY 

closed,  or  opened  anew,)  would  cause  the  magnetic  or  gal- 
vanic fluid  to  become  somewhat  varied  in  its  direction;  and 
thereby  cause  a  corresponding  variation  in  the  magnetic  needle. 
And  should  the  magnetic  meridians  be  found  to  correspond 
with  the  thicker  or  thinner  parts  of  the  crust,  or  with  lines  having 
reference  to  the  great  chains  of  volcanic  vents,  or  to  their  greater 
or  less  activity;  in  such  case,  the  declination  of  the  needle 
would  only  serve  to  indicate  where  those  sections  were. 

There  are  certainly  strong  indications  that  the  magnetic  (or 
galvanic)  current,  is  in  the  direction  from  the  equator  (or  rather 
the  magnetic  equator)  of  the  earth,  towards  the  external  poles  of 
the  earth  ;  and  that  in  all  this  distance,  at  least  until  we  arrive 
at  a  high  latitude,  it  is  received  and  conducted  by  the  surface 
of  the  earth,  in  a  curve  corresponding  with  the  surface  or  nearly 
so  ;  increasing  in  quantity  or  intensity  as  we  proceed  towards  the 
higher  latitudes ;  until  the  whole  shall  at  length  be  conveyed,  by 
this  conducting  curve,  to  the  internal  poles  ;  and  hence,  from 
this  constant  accumulation  in  quantity,  as  we  advance  towards 
the  higher  latitudes,  the  accepted  pole  of  the  needle  becomes 
more  strongly  attracted  or  more  strongly  inclined  to  correspond 
with  the  magnetic  current ;  and  hence,  the  oscillations  of  the 
needle  in  selling  to  its  meridian  will  be  more  rapid  as  we 
approach  the  higher  latitudes,  than  they  are  at  the  magnetic 
equator. 

But  it  is  by  no  means  certain  that  the  magnetic  fluid  becomes 
more  abundant  or  intense  up  to  90  degrees  of  latitude ;  and  if 
not,  (and  so  the  inference  would  seem  to  be,)  then,  whenever  the 
point  of  greatest  intensity  was  arrived  at,  the  dip  or  inclination 
of  the  needle,  would  be  at  90  degrees,  having  lost  all  horizontal 
traverse  or  direction,  and  thence  would  remain  in  that  situation 
or  nearly  so,  up  to  90  degrees  of  latitude;  and  hence,  the  com- 
pass-needle might  become  quite  useless  as  a  guide  to  the 
mariner  in  the  highest  latitudes. 

But  as  the  poles  of  magnetism,  electricity,  and  galvanism, 
(which  in  their  effects  and  operations  appear  to  be  but  different 
modifications  of  the  same  agent  or  power,)  are  so  inverse  in 
their  apparent  operations,  it  behoves  us  to  have  a  care,  lest  we 
assign  to  them  effects  that  do  not  belong  to  them.  Thus  it  is 
said,  "what  we  call  the  north  pole  of  the  magnet,  because 
attracted  by  the  north  pole  of  the  earth,  is,  for  this  very  reason, 
the  south  pole  of  the  magnet." 

Now  the  ground  of  this  dictum,  is  the  assumption  that 
the  external  poles  of  the  earth  are  the  only  actual  and  effec- 
tive poles  of  the  magnetic  or  galvanic  fluid  ;  and  if  such  were 
the  fact,  the  allegation  in  respect  to  the  compass-needle  or  magnet, 


OP     THE    SOLAR    SYSTEM.  243 

would  be  correct,  could  we  but  reconcile  the  idea  with  the  dip 
or  inclination  of  the  needle.  But  aside  of  the  little  consequence 
that  we  are  able  to  deduce  from  the  external  poles  of  the  earth, 
(except  that  of  giving  direction  to  the  compass-needle,  and  which, 
by  the  way,  becomes  less  and  less  a  direction  as  we  advance 
into  the  high  latitudes,  antipodes  as  those  external  poles  are  to 
each  other,  —  and  so  contrary  to  all  experience,  in  producing 
effects  by  the  adverse  poles  of  the  magnetic,  electric,  or  galvanic 
fluids;)  the  inferences,  would  seem  almost  unequivocal,  that 
the  fluid  of  what  we  commonly  term  the  north  pole  of  the 
magnet,  is  identical  with  that  at  the  external  north  pole  of 
the  earth  ;  and  that  the  error  has  arisen  from  supposing  the 
external  poles  of  the  earth  to  be  the  actual  poles  of  the  fluid ; 
and  not  that  the  earth  formed  a  conducting  circle  from  its  sur- 
face to  its  centre ;  and  that  the  pole  which  has  been  conceived 
to  have  its  direction  towards  the  north,  has  in  fact,  its  direction 
towards  the  south,  at  the  centre  of  the  earth,  where  it  meets  the 
opposite  pole,  and  where  the  battery  performs  its  operations. 

And  hence,  the  fluid  of  either  pole  of  the  magnet,  or  needle, 
is  always  impelled  in  the  line  of  direction  of  its  own  identical 
fluid;  seeking  connection  or  contact  with  the  opposite  pole  at 
the  centre  of  the  earth ;  but  when  fixed,  chained  and  iron  bound 
to  a  steel  bar,  and  not  at  liberty  to  pursue  its  wonted  course,  it 
is  always  ready  to  turn  aside,  and  enjoy  connection  with  the 
adverse  fluid,  whenever  opportunity  or  inducement  may  offer ; 
but  such  does  not  offer  when  fastened  at  opposite  ends  of  the 
same  straight  bar;  —  and  those  subtle  fluids  never  mistake  in 
their  object. 

I  will  then  assume  for  the  fluid  of  the  respective  poles  of  the 
magnet,  its  wonted  inclination  in  fact ;  viz.  that  of  flowing  in 
the  current  of  its  own  identity  when  left  free  to  do  so ;  and 
hence,  in  the  direction  assumed  by  the  compass  needle. 

I  have  said  that  the  adverse  currents  flow  all  the  way  from  the 
magnetic  equator  towards  the  external  poles  of  the  earth ;  and 
thence  to  their  respective  poles  at  the  centre.  This  was  only 
by  way  of  exemplification ;  for  there  is  much  evidence  to  show 
(by  way  of  inference  at  least)  that  the  whole  surface  of  the  earth 
is  both  a  receiver  and  conductor  of  the  fluids  for  both  poles  ;  but 
that  each  end  of  the  earth  only  conducts  its  respective  fluid  to  the 
centre,  or  central  poles.  Thus,  on  all  parts  of  the  earth,  so  far 
as  we  know,  artificial  magnets  can  be  formed  possessing  the 
adverse  fluids,  each  being  inclined  to  pass  with  the  fluid  of 
which  it  is  identical. 

If  this  theory  in  respect  to  magnetism  and  the  earth,  be  true  ; 
if  it  be  sustained  by  all  the  analogies  arising  from  experiments 


244  OUTLINE    OF    A   THEORY 

made  with  magnetism,  electricity  and  galvanism,  either  singly 
or  combined ;  if  it  be  impossible  otherwise  to  rationally  account 
for  many  of  the  phenomena  observed  in  geology,  and  especially 
that  hypothesis  so  strongly  inferred  as  to  compel  most  people  to 
implicitly  believe  the  hypothesis  a  fact,  and  to  endeavor  to  ac- 
count for  it  by  the  most  vague,  absurd,  and  unknown  cause, 
viz.  the  central  fires  of  the  earth  ;  —  then  let  us  adopt  the  theory, 
and  transfer  all  the  analogies  to  the  other  primary  planets,  and 
to  the  sun ;  all  of  which,  being  governed  by  the  same  great  laws 
of  nature  as  the  earth  is,  especially  by  that  of  attraction,  or  the 
great  physical  law  of  the  universe,  apparently  require  internal 
fires  as  much  as  the  earth  does ;  and  also  that  they  should  be 
made  in  the  same  way ;  and  if  beneficial,  that  they  should  be 
continued,  and  not  liable  to  be  extinguished,  nor  to  frighten  those 
mortals  who  inhabit  the  surface,  nor  put  them  in  danger  of  fall- 
ing into  them  by  breaking  through  the  crust. 

Conceive  then  that  the  sun  is  the  centre  of  our  system  :  and 
that  it  is  surrounded,  to  an  unknown  extent,  by  what  we  must 
probably  conceive  to  be  a  highly  attenuated  and  subtle  fluid, 
which  is  the  medium  of  attraction ;  and  that  this  fluid,  or 
medium,  becomes  less  dense  or  less  intense  in  proportion  to  the 
distance,  or  as  we  recede  from  the  sun ; 

That  by  and  through  this  medium  of  attraction,  the  sun  be- 
comes the  magnetizer  of  all  gross  matter  in  the  solar  system ;  in 
which  case,  the  primary  planets  are  magnetized  in  the  propor- 
tion to  their  inverse  distance  from  the  sun ;  thereby  giving  them 
attractive  powers  in  proportion  to  their  bulks  directly  and  their 
inverse  distance  conjointly ;  and  hence  their  inverse  distances 
from  the  sun  denote  their  respective  rates  of  gravity  towards 
the  sun  ; 

That  each  planet,  from  being  thus  magnetized  by  the  sun,  be- 
comes itself  imbued  by  a  portion  of  the  magnetic  fluid,  the 
density  of  which  is  proportioned  to  the  bulk  of  the  planet  direct- 
ly and  its  inverse  distance  from  the  sun  conjointly;  and  hence, 
its  power  to  attract  other  matter ;  and  in  such  case,  it  would 
be  hard  to  conceive  that  the  magnetic  atmosphere  of  the  sun 
would  be  a  resisting  medium  to  the  planet,  from  the  circum- 
stance that  the  fluid  as  readily  attaches  to  the  planet  as  to  the 
sun ;  in  which  case,  all  resistance  to  the  motion  of  the  planet 
would  be  annihilated  far  in  advance  ; 

That  the  sun  and  planets  are  supplied  with  internal  fires  in 
the  same  manner  that  the  earth  is ;  and  that  too  in  proportion  to 
their  respective  surfaces  and  their  powers  of  attraction  ;  and  also 
with  volcanic  flues  proportional,  as  outlets  to  such  internal  fires ; 

That  neither  light  nor  attraction,  or  gravitation,   are   emana- 


OF     THE    SOLAR    SYSTEM.  245 

tions  from  matter,  in  the  sense  in  which  they  have  generally 
been  considered  ;  that  neither  of  them  are  progressive  in  time; 
but  are  instantaneous  in  their  effects  at  all  distance  subject  to 
their  effects  ;  arid  hence,  that  the  effect,  or  rate  of  intensity  of 
each,  is  inversely  as  the  distance;  nor  has  light  anything  to  do 
with  filling  space  with  particles,  or  rays  of  light,  as  has  in  gene- 
ral been  taught  and  understood; 

That  the  atmospheres  of  the  sun  and  planets,  properly  so 
called,  (viz.  that  which  consists  of  the  gasses,  gross  vapors,  and 
impalpable  matter  that  surrounds  those  bodies,)  may  well  be 
conceived  to  have  some  near  proportion  to  the  size,  and  to  the 
power  of  attraction  of  the  bodies  which  they  surround;  the  den- 
sity of  such  atmosphere  depending  upon  its  elasticity  and  the 
power  by  which  it  is  attracted,  must  necessarily  depend  on  the 
amount  of  matter  which  it  contains,  and  the  power  of  attraction 
with  which  it  is  drawn  towards  the  body  which  it  surrounds ; 
and  hence  its  capability  to  affect  light  by  refraction,  reflection  or 
otherwise ;  and  hence  its  power  for  raising  gross  vapors  from  the 
body  which  it  surrounds ;  which  in  case  of  the  sun,  must  be 
with  great  rapidity,  and  to  a  great  height; 

That  when  we  view  the  sun  through  a  telescope,  we  see  its 
solid  parts  ;  but  through  an  extremely  dense  and  brilliant  atmos- 
phere of  vast  extent ; 

That  the  spots  often  observed  upon  the  sun  are  caused  by 
volcanic  eruptions,  or  discharges  of  fire,  smoke,  vapors  and  oth- 
er substances  from  the  vents  to  the  internal  fires  ;  which,  rising 
in  vast  quantities,  with  great  rapidity,  and  to  a  great  height,  often 
spread  the  umbra  to  a  vast  extent,  so  as  to  hide  large  portions  of 
the  sun's  surface  from  our  view ;  and  while  the  eruption  con- 
tinues, the  boundary  between  the  nucleus,  or  matter  immediately 
ejected,  and  the  umbra,  will  be  well  defined;  and  also,  that  the 
nucleus  will  naturally  vanish  before  the  umbra ;  and  on  their 
both  vanishing,  a  red  hot  lava  may  for  a  time  present  to  view 
what  is  termed  a  facula,  until  it  shall  at  length  take  the  appear- 
ance of  other  parts  of  the  surface  of  the  sun. 

If,  however,  the  sun,  or  centre  of  the  solar  system,  be  sur- 
rounded by  a  vast  magnetic  atmosphere  of  ever  so  great  tenuity, 
if  the  same  vary  in  density  in  the  inverse  ratio  of  the  distance, 
and  is  sufficiently  dense  to  refract  light,  (the  central  battery  of 
which  fluid  is  at  the  centre  of  the  sun,)  the  same  would  most  ad- 
mirably account  for  the  phenomena  observed  by  Dr.  Bradley,  from 
which  he  deduced  his  hypothesis  of  the  progressive  motion  of 
light  in  time,  as  also  for  the  zodiacal  light.  Notwithstanding, 
La  Place  would  reject  the  last  suggestion,  for  the  reason  that  it 
would  interfere  with  the  Newtonian  theory  of  gravity,  which  he 
has  so  liberally  endorsed  to  the  world. 


CHAPTER    VI. 

On  the  Tides. 

IN  accounting  for  the  tides  upon  the  principle  of  the  attractive 
powers  of  the  sun  and  moon  upon  the  earth,  and  the  waters  of 
the  earth ;  those  who  have  attempted  it,  have  assumed  as 
a  lemma,  or  at  least  an  hypothesis,  that  the  force  of  gravity 
is  inversely  as  the  square  of  the  distance  ;  and  as  the  appa- 
rent diameter  of  the  moon  is  to  that  of  the  sun  when  viewed 
from  the  earth,  as  1.091  to  1,  and  as  according  to  their  lav/ 
of  gravity,  the  respective  influences  of  the  sun  and  moon  upon 
the  earth,  should  be  as  their  apparent  diameters,  in  case  their 
densities  were  equal,  hence,  according  to  their  apparent  diam- 
eters, the  moon  would  have  but  very  little  more  influence  upon 
the  tides  than  the  sun  has. 

But  finding  that  the  greater  part  of  the  tides  are  caused  by  the 
influence  of  the  moon,  it  has  been  concluded  that  the  moon  is 
much  more  dense  than  the  sun ;  which  densities  as  proportioned 
to  each  other,  they  have  weighed  and  determined,  so  far  at  least 
as  to  satisfy  themselves  and  many  others,  by  resorting  to  a 
comparison  of  their  supposed  respective  influences  upon  the 
tides ;  which,  to  be  sure,  has  cost  them  no  small  trouble ;  nor 
have  their  conclusions  been  very  uniform  or  satisfactory,  even 
to  themselves. 

Bernoulli  estimates  the  influence  of  the  moon  to  that  of  the 
sun  upon  the  tides,  as  2.5  to  1 ;  Newton,  as  3.5  to  1 ;  and  others 
have  even  estimated  it  as  1  6-7th  to  1 ;  La  Lande,  as  2  7-lOths 
to  1 ;  and  La  Place  3  to  1.  But  Bernoulli's  ratio  is  in  general 
settled  upon,  as  being  thought  the  most  probable.  Hence  the 
conclusion,  that  if  the  joint  influence  of  the  sun  and  moon  will 
raise  the  tides  7.445  feet,  the  moon's  portion  will  be  5.412  feet, 
while  that  of  the  sun  will  be  2.033  feet. 

But  after  all,  they  seem  to  labor  under  difficulties  in  adapting 
their  theory  to  observation ;  for  Mr.  Vince,  after  enumerating 
many  difficulties  of  this  kind,  observes,  "  Thus  the  theory  alone 
will  afford  no  practical  conclusions."  And  in  justification  of 


ON    THE    TIDES.  '    .       247 

certain  tables  based  or  founded  on  theory, — after  consenting  that 
they  will  not  agree  with  observation,  —  says :  "  To  determine, 
therefore,  the  true  time  at  any  port,  we  must  find  from  observa- 
tion what  is  the  difference  between  the  true  time  and  that  shown 
by  the  table,  and  then  'that  difference  added  to  the  time  shown 
by  the  table  will  give  the  time  of  the  high  tide." 

In  the  American  Encyclopaedia  or  Conversations  Lexicon,  the 
reader,  under  the  article  Tides,  will  find  a  somewhat  racy  expo- 
sition of  the  commonly  received  theory  of  the  tides,  and  as  the 
articles  in  that  work,  though  often  giving  but  an  epitome  of  the 
subjects  of  which  they  treat,  are  nevertheless  written  with  ability, 
giving  the  most  approved  philosophy  of  the  day  in  which  they 
were  written,  —  as  is  the  case  with  the  article  alluded  to,  which 
is  an  article  also  well  calculated  to  satisfy  the  minds  of  most 
people  upon  a  subject  of  which  they  had  thought  and  cared  but 
little,  and  who  were  not  very  curious  to  know  if  these  things 
were  so,  —  and  as  it  is  somewhat  convenient  to  have  it  here 
inserted,  with  a  view  fo  commentary  and  comparison,  I  will 
make  a  liberal  extract  from  the  same,  notwithstanding  the  work 
itself  is  readily  accessible  to  most  readers. 

"  The  ebb  and  flow  of  the  sea  are  evidently  connected  with 
the  moon's  motions.  The  level  of  the  ocean  is  slightly  disturbed 
by  the  attraction  which  is  alternately  exerted  and  withdrawn. 
The  waters  for  a  large  space  under  the  moon,  being  more  at- 
tracted than  the  great  body  of  the  earth,  are  thus  rendered  light- 
er than  those  parts  of  the  ocean  which  are  at  the  same  distance 
as  the  earth's  centre,  and  being  lighter,  they  are  forced  upwards 
a  little  by  the  surrounding  mass,  which  is  heavier,  just  as  water 
and  oil  will  stand  at  different  heights  in  the  two  branches  of  a 
syphon  tube  ;  or  just  as  ice,  which  is  lighter  than  water,  is  made 
to  rise  a  little  higher  on  that  account,  when  placed  in  water. 

"  If  the  earth  rested  immovable  upon  a  fixed  support,  there 
would  be  a  tide  or  rising  of  the  waters  only  on  the  side  towards 
the  moon.  But  the  great  body  of  the  earth  is  just  as  free  to 
move  as  a  single  particle  of  the  ocean,  and  if  suffered  to  yield 
to  the  moon's  attraction,  would  be  carried  just  as  fast.  Hence 
for  the  same  reason  that  a  particle  of  water,  on  the  side  of  the 
earth  towards  the  moon,  is  drawn  away  from  the  centre,  or  has 
its  downward  tendency  diminished,  so  the  solid  earth  itself  is 
drawn  away  from  the  mass  of  waters  on  the  side  of  the  earth 
farthest  from  the  moon.  It  is  the  difference  of  attraction  in  both 
cases  between  the  surface  and  the  centre,  which  causes  the  light- 
ness of  the  waters  and  the  consequent  elevation.  It  will  be 
seen,  therefore,  that,  taking  the  whole  earth  into  view,  there  are 
always  two  high  tides  diametrically  opposite  to  each  other,  and 


248  ON    THE    TIDES. 

two  low  tides  also  midway  between  the  high  ones.  The  high 
tides  are  two  great  waves,  or  swells,  of  small  height,  but  extend- 
ing each  way  through  half  a  right  angle.  These  waves  follow 
the  moon  in  its  monthly  motion  round  the  earth,  while  the  earth, 
turning  on  its  axis,  causes  any  given  place  to  pass  through  each 
of  these  swells  and  the  intervening  depressions  in  a  lunar  day, 
or  twenty-four  hours  and  fifty  minutes.  What  we  have  said  with 
respect  to  the  moon's  influence  in  disturbing  the  level  of  the 
ocean,  may  be  also  applied  to  that  of  the  sun ;  only  in  the  case 
of  the  sun,  although  its  absolute  action  is  about  double  that  of 
the  moon,  yet  on  account  of  its  very  great  distance,  its  relative 
action  upon  the  surface  of  the  earth  compared  with  that  at  the 
centre,  is  but  about  one  third  as  great  as  that  of  the  moon.  At 
the  new  and  full  moon,  when  the  sun's  and  moon's  action  con- 
spires, the  tides  are  highest,  and  are  called  spring  tides.  But  at 
the  first  and  last  quarters  of  the  moon,  the  action  of  one  body 
tends  to  counteract  that  of  the  other;  and  the  tides,  both  at  ebb 
and  flow,  are  called  neap  tides."  "  That  the  high  tide  when  the 
moon  is  above  the  horizon,  exceeds  the  high  tide  when  the  moon 
is  below  the  horizon.  And  that  50°  towards  the  nearest  pole 
from  where  the  moon  is  vertical,  there  will  be  only  one  tide  in 
twenty-four  hours,"  &c. 

Now  the  theory  contained  in  this  running  account  of  the 
phenomena  of  the  tides,  (though  it  were  the  result  of  La  Place's 
greatest  intellectual  effort,)  is  not  calculated  to  give  full  satisfac- 
tion upon  the  subject. 

1st.  The  waters  on  the  side  of  the  earth  towards  the  moon, 
are  more  attracted  than  the  great  body  of  the  earth,  —  made 
lighter  than  the  waters  at  the  distance  of  the  centre  of  the  earth, 
—  and  hence  forced  up  by  the  surrounding  mass  of  waters. 

2d.  If  the  earth  were  made  immovable,  (viz.,  so  that  its  centre 
and  solid  parts  could  not  be  attracted  towards  the  moon,)  there 
would  be  a  tide  only  on  the  side  of  the  earth  towards  the  moon. 
But  as  it  is  not  fixed,  the  solid  earth  is  drawn  from  the  mass  of 
waters  on  the  side  of  the  earth  opposite  the  moon,  and  hence, 
creating  a  tide  on  the  side  of  the  earth  opposite  the  moon. 

3d.  That  the  tides  on  both  sides  of  the  earth  are  caused  by 
the  difference  of  attraction  between  the  surface  and  centre  of  the 
earth. 

4th.  That  the  high  tides  are  two  great  waves  or  swells,  ex- 
tending each  way  through  half  a  right  angle,  following  the  moon 
in  its  monthly  motion  round  the  earth. 

5th.  That  the  sun  has  a  like  influence  upon  the  waters  that 
the  moon  has,  —  only,  though  the  sun  has  about  twice  the  abso- 
lute action  upon  the  earth  that  the  moon  has,  yet  from  its  great 


ON    THE    TIDES.  249 

distance,  its  relative  action  upon  the  surface  of  the  earth  com- 
pared with  that  at  the  centre,  is  but  about  one  third  as  great  as 
that  of  the  moon. 

In  the  foregoing  theory  there  is  certainly  an  ample  number  of 
causes  assigned  to  produce  the  effect  or  phenomena,  in  case  they 
are  such  as  are  sufficient.  Thus,  if  the  attraction  of  the  moon 
makes  the  waters  on  the  side  of  the  earth  towards  the  moon 
lighter  than  they  are  at  the  distance  of  the  centre  of  the  earth  from 
the  moon,  so  that  they  are  forced  upward  by  the  heavier  sur- 
rounding waters,  this  cause  would  be  sufficient  for  causing  the 
tide  on  the  side  of  the  earth  towards  the  moon  ;  and  would 
seem  to  dispense  with  that  of  their  being  attracted  away  from 
the  central  parts  of  the  earth.  But  as  the  surrounding  waters 
could  only  press  those  upward  which  were  made  lighter  on  that 
side  of  the  earth  towards  the  moon,  this  cause  alone  could  not, 
therefore,  account  for  the  phenomena  of  the  opposite  tide  ;  and 
hence,  inasmuch  as  the  opposite  tide  must  be  accounted  for 
from  some  other  cause,  and  by  one  which  would  as  well  account 
for  both  as  for  either,  the  whole  cause  seems  rather  at  length  to 
be  assigned  to  that  of  the  waters  on  the  side  of  the  earth  towards 
the  moon  being  attracted  from  the  central  parts  of  the  earth ; 
and  the  central  parts  of  the  earth  being  attracted  and  drawn 
from  the  waters  on  the  opposite  side  of  the  earth. 

This  doubtless  goes  upon  the  ground,  that  if  the  earth  were 
an  entire  mass  of  waters,  and  were  not  disturbed  by  any  foreign 
power,  the  mass  \vould  be  in  shape  of  a  perfect  sphere ;  but  if 
attracted  by  the  moon,  (for  instance,)  it  would  become  elongated; 
its  greatest  diameter  being  directed  towards  the  moon,  — in  like 
manner  as  Sir  Isaac  Newton  conceives  the  moon  to  be  in 
respect  to  the  earth ;  with  this  difference,  however,  that  while 
Newton  supposed  the  end  of  the  moon  which  hung  next  to  the 
earth  was  heavier  than  the  other  end,  the  contrary  would  be  the 
case  in  respect  to  the  ends  of  the  earth. 

But  we  will  suppose  the  internal  parts  of  the  earth  to  be  solid, 
or  composed  of  solid  matter,  having  its  surface  covered  with 
deep  waters ;  and  then  suppose  the  tides  to  be  two  broad  waves 
directly  opposite  each  other ;  that  the  one  nearest  the  moon  shall 
be  caused  by  being  attracted  from  the  solid  parts  of  the  earth, 
and  the  opposite  one  shall  be  caused  by  the  solid  parts  of  the 
earth  being  attracted  from  the  opposite  waters  ;  the  high  water  of 
that  wave  nearest  the  moon  being,  however,  some  three  hours 
or  45°  behind  the  moon,  (or  the  position  of  the  moon,  caused 
by  the  diurnal  motion  of  the  earth ;)  and  then  let  us  endeavor  to 
calculate  the  motion  of  the  solid  parts  of  the  earth,  (as  well  as  of 
32 


250  ON    THE    TIDES. 

those  two  great  waves)  caused  by  the  disturbing  force  of  the 
moon. 

Will  it  be  said  in  such  case,  that  the  solid  parts  of  the  earth, 
in  consequence  of  the  attraction  of  the  moon,  make  or  perform 
a  revolution  in  an  orbit,  in  about  twenty-four  hours  and  fifty 
minutes;  the  direction  of  which  orbit  will  be  subject  also  to  all 
the  wanderings,  or  northings  and  southings  of  the  moon;  and 
that  the  direction  of  the  solid  parts  of  the  earth  in  such  orbit  is 
always  towards  the  point  of  high  tide,  in  that  wave  or  swell 
nearest  the  moon  ? 

If  so,  to  calculate  and  adjust  the  equations  of  the  earth's  an- 
nual orbit,  required  in  consequence  of  this  disturbing  force,  would 
puzzle  even  a  La  Place. 

But  it  is  said  the  sun  has  about  twice  the  influence  upon  the 
earth  that  the  moon  has,  but  from  its  very  great  distance,  its  rela- 
tive influence  upon  the  surface  of  the  earth,  is  but  about  one 
third  as  great  as  that  of  the  moon  compared  with  that  at  the 
centre.  Hence,  according  to  this  hypothesis,  the  influence  of 
the  sun  upon  the  centre  of  the  earth,  must  be  vastly  greater 
than  that  of  the  moon  ;  and  I  will  here  merely  query,  how 
those  joint,  opposing,  or  varied  influences  of  the  sun  and  moon, 
are  to  affect  the  centre  and  surface  of  the  earth,  in  all  the  wan- 
derings of  the  moon;  how  in  conjunction,  in  opposition,  in 
quadratures,  &c. 

To  me,  the  whole  theory  is  but  a  tissue  of  absurdities  and 
dilemmas ;  as  though  cause  and  effect  had  been  guessed  off, 
without  much  consideration  with  regard  to  their  connection, 
mutuality,  or  reciprocity;  and  as  if  the  connecting  links  had 
been  buf  little  regarded. 

Nor  does  the  foregoing  theory  differ  essentially  from  that  of 
Mr.  Vince.  Mr.  Vince, —  after  observing  that  Kepler  was  the 
first  who  assigned  the  true  physical  cause  of  the  tides,  namely, 
"  that  the  waters  of  the  sea  gravitate  towards  the  moon,  and 
cause  the  tides;"  and  that  Sir  Isaac  Newton  had  shown  that 
from  the  principles  of  gravity,  the  phenomena  of  the  tides  may 
be  solved,  —  proceeds  with  an  attempted  exposition  of  the  pheno- 
mena, upon  the  Newtonian  principles,  by  referring  to  the  sup- 
posed operation  of  corpuscles,  of  infinitely  small  magnitudes,  up- 
on each  other,  (by  which  the  world  has  been  so  much  darkened 
in  its  researches  after  truth ;)  and  after  alleging  difficulties  that 
constantly  prevent  fact  from  agreeing  with  theory,  as  for  instance, 
"  that  the  free  motion  of  the  waters  in  the  open  seas  is  hindered 
by  shallow  places,  rocks,  and  islands,  in  consequence  of  which, 
the  tides  in  some  of  the  open  seas,  at  the  time  of  the  conjunction 
of  the  luminaries,  are  found  to  rise  only  to  the  height  of  about 


ON    THE    TIDES.  251 

three  feet,5'  proceeds  to  remark  thus,  in  which  the  case  seems 
rather  to  be  summed  up : 

"  The  general  phenomena  of  the  tides  from  observation, 
agree  very  well  with  the  conclusions  deduced  from  the  theory  of 
gravity,  indeed  much  more  accurately  than  could  have  been 
expected,  when  we  consider  how  many  circumstances  there  are 
which  take  place,  and  which  cannot  be  reduced  to  computation. 
The  theory  supposes  the  wrhole  surface  of  the  earth  to  be  covered 
with  deep  waters,  that  there  is  no  inertia  of  the  waters,  that 
the  major  axis  of  the  spheroid  is  constantly  directed  to  the  moon, 
and  that  there  is  an  equilibrium  of  all  the  parts.  But  the  inertia 
of  the  waters  will  make  them  continue  to  rise  after  they  have 
passed  the  moon,  although  the  action  of  the  moon  begins  to 
decrease,  and  they  come  to  their  greatest  altitude  in  the  open 
seas  about  three  hours  after,  at  which  time  there  is  not  a  general 
equilibrium,  but  the  waters  rise  and  fall  by  a  reciprocation  ; 
hence,  the  longest  axis  is  not  directed  to  the  moon,  nor  is  the  figure 
a  perfect  spheroid.  The  waters  have  not  a  free  motion  on  account 
of  the  shallow  places,  rocks,  islands,  and  continents,  the  force  of 
currents  and  winds;  also  as  the  waters  approach  the  equator 
where  the  earth  has  a  greater  velocity  about  its  axis,  they  must 
necessarily  be  left  behind,  and  obstruct  the  regular  motion  of  the 
water  when  it  moves  from  west  to  east,  but  conspire  with  that 
from  east  to  west. 

"  All  these  circumstances  must  affect  the  measures  of  the  phe- 
nomena as  deduced  from  theory ;  it  may,  however,  in  many  cases, 
give  the  relative  measures  without  any  great  error,  so  that  by 
accurate  observations  once  made,  on  their  absolute  quantity,  in 
some  one  particular  case,  the  measures,  in  all  other  cases,  may 
be  ascertained  to  a  considerable  degree  of  accuracy." 

Now,  from  the  phenomena  of  the  tides,  as  found  from  obser- 
vation, perhaps  no  one  would  doubt  Kepler's  hypothesis,  that 
they  were  caused  by  the  influence  of  the  moon ;  and  had  not  New- 
ton and  Bernoulli  subterfuged  and  perplexed  the  theory,  in 
Newton's  mysterious  law  of  gravity,  in  connection  with  the  ease 
and  facility  with  which  La  Place  has  been  able  to  equate  the 
effects  of  all  disturbing  forces  that  might  be  suggested,  it  is 
highly  probable  that  the  world,  from  observation,  would  long 
ago,  have  come  to  more  rational  conclusions  as  to  the  manner  in 
which  the  influence  of  the  moon  is  exerted  in  causing  the  tides. 

Mr.  Vince,  who  always  treats  general  subjects  with  the  utmost 
candor,  seems  rather  cautiously  to  have  avoided  descanting 
much  upon  the  cause  of  the  tide  on  the  side  of  the  earth  oppo- 
site the  moon  ;  and  hence,  has  left  this  part  of  the  subject  rather 
destitute  of  explanation. 

I  have  thus  presented  some  of  the  palpable  and  insurmount- 


252  ON    THE    TIDES. 

able  difficulties  under  which  the  popular  theories  labor  in  at- 
tempting to  account  for  the  phenomena  of  the  tides  upon  the 
Newtonian  law  of  gravity,  and  more  especially  upon  the  New- 
tonian hypothesis  of  universal  gravity,  which,  in  my  humble 
opinion,  lays  the  whole  foundation  of  the  fabric  of  astronomy 
in  fortuity  and  atheism.  And  if.  as  is  so  whimsically  alleged,  for 
the  purpose  of  giving  it  a  marvellous  effect,  this  hypothesis  was 
the  offspring  of  the  fall  of  an  apple,  the  consequences  of  that  fall, 
(comparing  small  things  with  great,)  were  not  less  deleterious  to 
the  scientific  world,  than  was  the  forbidden  fruit  to  the  moral 
world. 

I  will  now,  after  a  few  preliminary  remarks  upon  the  law 
of  gravitation  according  to  my  determination,  (viz.,  that  the 
force  of  gravity  varies  inversely  as  the  distance  varies,)  give 
an  outline  of  my  theory  of  the  tides,  based  upon  my  theory  of 
universal  gravity  as  connected  with  my  theory  of  the  solar  system. 

By  rny  determination  of  the  law  of  gravity,  if  the  moon  were 
to  revolve  about  the  earth  in  an  orbit  but  l-4th  the  distance  that 
it  now  does,  its  influence  upon  the  earth  would  be  four  times  what 
it  now  is,  and  by  the  Newtonian  law  of  gravity,  its  influence 
would  be  sixteen  times  what  it  now  is.  But  according  to  my  the- 
ory of  the  tides,  I  should,  according  to  my  law  of  gravity,  make  its 
influence  upon  the  tides  a  little  more  than  four  times  what  it  now 
is ;  and  by  the  Newtonian  law  of  gravity,  I  should  make  its  in- 
fluence upon  the  tides  a  little  more  than  sixteen  times  what  it  now 
is ;  for  the  motion  of  the  moon  in  its  orbit  must  of  necessity,  be 
just  double  w^hat  it  now  is,  if  it  revolve  four  times  as  near  the 
earth ;  and  hence,  the  motion  of  the  moon  in  its  orbit  being 
doubled,  its  apparent  motion  over  the  surface  of  the  earth,  caused 
by  the  diurnal  motion  of  the  earth  upon  its  axis,  would  not  be 
quite  as  great  as  it  now  is.  But  whether  upon  the  popular 
theories  of  the  tides,  what  are  conceived  to  be  the  great  tidal 
waves  of  the  ocean  would  swell  to  four  times  the  height  in  the 
one  case,  or  to  sixteen  times  the  height,  in  the  other  case,  that  they 
are  now  conceived  to  swell,  I  will  not  undertake  to  determine. 

The  estimated  bulk  or  magnitude  of  the  moon  to  that  of  the 
earth  (which  is  doubtless  correct)  is  as  1  to  49,  and  if  their 
influences  or  attractive  powers  upon  each  other  are  in  propor- 
tion to  their  respective  bulks  or  magnitudes,  (and  from  all  anal- 
ogy as  yet  deduced  from  observation,  there  is  little  or  no  doubt 
of  the  fact,)  their  respective  influences  upon  each  other  will  be 
as  1  to  49. 

If  we  now  take  round  numbers,  and  consider  the  distance  of 
the  moon  from  the  earth  to  that  of  the  earth  from  the  sun  as  in  the 
proportion  of  1  to  400, —  which  is  in  accordance  with  the  estima- 
tion of  some,  and  is  sufficiently  near  for  our  present  purposes, 


ON    THE    TIDES.  253 

although  it  makes  the  distance  of  the  moon  somewhat  less 
than  the  common  estimate,  if  we  call  the  distance  of  the  sun 
95,000,000  miles  from  the  earth, —  in  such  case,  according  to 
my  determination  of  the  law  of  gravity,  the  attractive  power  of 
the  sun  upon  the  earth  or  moon  would  be  some  eight  hundred 
and  ninety-live  times  as  great  as  that  of  the  attractive  power  of 
the  earth  upon  the  moon  ;  as  denoted  and  determined  by  the  mo- 
tion of  the  earth  in  its  orbit  about  the  sun,  and  the  motion  of  the 
moon  in  its  orbit  about  the  earth ;  and  consequently,  that  the  at- 
tractive power  of  the  sun  upon  the  earth  would  be  more  than 
forty  thousand  times  as  great  as  that  of  the  moon  upon  the 
earth. 

And,  yet,  according  to  my  theory  of  the  tides,  no  portion  of 
them  are  caused  by  the  direct  influence  of  the  sun  upon  the 
earth,  or  upon  its  waters ;  although  its  indirect  influence  exerted 
through  the  moon,  causes  the  variation  from  spring"  to  neap  tides. 
And  in  fact,  if  my  determination  of  the  law  of  gravity  be 
correct,  it  is  quite  certain  that  the  influence  of  the  sun  upon  the 
tides  compared  with  that  of  the  moon  (whether  direct  or  indi- 
rect) must  be  exceedingly  small,  when  compared  with  their 
respective  attractive  powers  upon  the  earth. 

But  I  will  conclude  my  investigations  of  the  theories  extant, 
in  the  words  of  Mr.  Vince,  in  which  he  again  sets  forth,  by  way 
of  repetition  or  recapitulation,  the  difficulties  to  be  encountered 
under  present  theories ;  and  will  thence  proceed  to  a  theory  of  my 
own.  After  many  attempted  explanations  of  the  causes  and 
phenomena  of  the  tides  upon  the  Newtonian  theory  of  gravity, 
he  says,  "  It  has  been  here  supposed,  that  the  high  tide  was 
under  the  luminary,  and  that  there  was  a  general  equilibrium  of 
the  waters  ;  but  the  high  tide  is  at  some  distance  from  the  lumi- 
nary, and  the  waters  rise  and  fall  by  a  reciprocation ;  also  the 
free  motion  of  the  waters  in  the  open  seas  is  hindered  by  shal- 
low places,  rocks  and  islands  ;  in  consequence  of  which,  the 
tides  in  some  of  the  open  seas,  at  the  time  of  the  conjunction  of 
the  luminaries,  are  found  to  rise  only  to  the  height  of  about 
three  feet,"  and  concludes  by  saying,  "  thus  the  theory  alone  will 
afford  no  practical  conclusions." 

So  much  for  Mr.  Vince,  and  other  authorities  upon  the  theory 
of  the  tides. 

I  will  now  present  a  short  outline  of  my  own  theory  of  the 
tides,  based  upon  my  theory  of  universal  gravitation,  and  upon 
my  theory  of  the  solar  system,  submitting  it  to  the  consideration 
of  an  inquiring  world,  who  have  ever  been  anxious,  from  the 
days  of  Aristotle  and  Virgil,  to  know 

Why  flowing  tides  prevail  upon  the  main, 
And  in  what  dark  recess  they  shrink  again. 


254  ON    THE    TIDES. 

Malte  Brun,  who  was  somewhat  of  a  whimsical  and  credu- 
lous philosopher,  (however  eminent  he  was  as  a  geographer,) 
treats  of  the  tides  mostly  after  the  popular  methods ;  he  has 
some  remarks,  however,  not  unworthy  of  notice,  as  "that  they 
have  tides  in  the  islands  of  the  South  Sea  of  only  one  or  two 
feet  elevation,  whilst  upon  the  western  coasts  of  Europe  and 
the  eastern  coasts  of  Asia  the  tides  are  extremely  strong  and 
subject  to  many  variations."  Now  Europe  and  Asia  make  a 
very  broad  continent ;  and  the  western  shores  of  Europe  and 
the  eastern  shores  of  Asia  are  washed  by  broad  oceans. 

The  same  author  says,  "  In  the  torrid  zone,  the  flood-tides  run 
from  east  to  west  with  the  motion  of  the  stars,"  but  he  says 
nothing  as  to  the  direction  of  the  ebb  of  such  tides,  and  might 
leave  us  to  infer  that  the  tides  in  the  torrid  zone  may  have  some 
connection  with  those  ocean  currents  with  which  he  has  made 
such  a  display,  that  it  has  been  thought  prudent  to  suggest,  that 
he  may  have  accounted  for  more  than  existed. 

Such  ocean  currents,  however,  as  are  known  to  exist,  and 
which  have  been  traced,  or  the  courses  of  which  have  been  ascer- 
tained, seem  very  well  calculated  to  aid  and  corroborate  a  true 
theory  of  the  tides;  nor  do  I  conceive  it  a  matter  so  inexplicable  as 
the  author  above  named  would  intimate,  that  a  current  should 
proceed  from  the  Ethiopian  ocean  to  the  coasts  of  Brazil,  —  that 
a  current  should  proceed  from  the  coast  of  Brazil  and  extend 
towards  the  eastern  coasts  of  America,  and  prevail  between 
30°  north,  and  10°  south  latitude ;  and  that  within  the  above 
limits  (of  the  degrees  of  latitude  no  doubt)  there  should  be  a  cur- 
rent running  the  contrary  way,  or  toward  the  coast  of  Africa. 
These  currents,  however,  all  seem  inclined  to  run  towards  conti- 
nents, or  large  tracts  of  land,  and  thence  to  float  along  the  shores 
or  near  them,  and  to  pass  strongly  through  straits,  or  narrow 
passes,  as  between  New  Holland  and  Van  Diemen's  Land, 
&c.  But  few  of  the  ocean  currents,  however,  have  been  very 
distinctly  traced ;  nor  will  I  do  more  than  merely  to  suggest  or 
hint  at  a  subject  so  little  known  or  understood,  in  conroboration 
of  my  theory  of  the  tides. 


SECTION    SECOND. THEORY    OF    THE    TIDES. 

My  theory  is,  that  the  tides  are  caused  by  the  direct  influence  of 
the  moon,  the  sun  exerting  no  other  influence  in  raising  the 
tides  than  an  indirect  one  exercised  through  the  moon ;  al- 
though the  attractive  influence  of  the  sun  upon  the  earth  is  many 
thousand  times  greater  than  that  of  the  moon  upon  the  earth ; 


THEORY    OF    THE    TIDES.  255 

That  the  earth  possesses  \wofixed  poles  in  respect  to  the  sun, 
which  are  susceptive  of,  and  obedient  to  solar  magnetism,  (or  the 
magnetic  fluid  of  the  solar  system,)  in  conducting  the  same  to  the 
centre  of  the  earth,  for  the  purposes  designed;  which  solar  mag- 
netism so  wholly  and  powerfully  pervades  the  whole  earth,  that 
solar  magnetism  is  not  perceptible  in  one  part  of  the  earth  more 
than  in  another,  if  we  except  the  stronger  influence  that  may 
exist  at  the  poles;  but  that  the  earth  has  no  movable  poles  in 
respect  to  the  sun  ; 

That  the  moon  has  but  one  fixed  pole  in  respect  to  the  earth, 
and  none  in  respect  to  the  sun  ;  but  that  its  poles  in  respect  to 
the  sun,  are  movable  and  revolving ;  one  always  being  towards 
the  sun,  and  the  other  opposite  to  the  sun.  And  that  similar  to 
this,  is  the  earth  in  respect  to  the  moon,  so  far  as  the  mag- 
netic influence  of  the  moon  is  capable  of  creating  polarity 
in  the  earth ;  the  poles  of  the  earth  created  by  the  moon  being 
movable,  or  revolving  poles,  one  always  being  directed  towards 
the  moon,  and  the  other  in  the  opposite  direction.  And  although 
those  poles  may  be  said  to  have  some  influence  over  the  whole 
of  both  hemispheres  of  the  earth,  yet  the  greatest  attractive 
power  of  those  poles  would  always  be  directly  underhand  op- 
posite to  the  moon,  at  which  points  matter  would  always  be 
inclined  to  accumulate. 

That  the  moon  has  but  one  fixed  pole  in  respect  to  the  earth, 
may  fairly  be  inferred  from  the  circumstance  that  one  and  the 
same  side  of  the  moon  is  always  retained  towards  the  earth.  A 
circumstance  which  Newton  attempted  to  account  for  upon 
the  hypothesis  that  the  moon  might  be  elongated,  and  that, 
consequently,  one  end  would  hang  towards  the  earth,  while 
others  have  as  gravely  supposed,  that  one  side  of  the  moon 
might  be  heavier  than  the  other.  If,  however,  our  present  con- 
ceptions of  polarity  and  magnetism  will  not  permit  the  idea  of 
a  fixed  or  permanent  pole  without  its  opposite,  there  surely  could 
be  no  objection  to  conceiving  the  moon  to  have  two  fixed  poles 
in  respect  to  the  earth,  both  of  them  situated  on  the  side  of  the 
moon  next  to  the  earth  (exercising  their  influence  in  and  through- 
out the  moon,)  any  more  than  to  a  magnet  having  both  its  poles 
in  one  direction. 

If  the  foregoing  hypothesis  in  respect  to  the  magnetic  or  at- 
tractive influences  of  the  earth,  sun,  and  moon,  upon  each  other, 
be  adopted,  the  cause  and  phenomena  of  the  tides  are  readily 
accounted  for. 

In  such  case,  the  movable  poles  of  the  earth  created  by  the 
moon,  would  be  constantly  tending  to  attract  matter  toward 
those  poles ;  and  should  we  conceive  the  earth  composed  of  a 
fluid,  as  of  water, —  or  to  be  covered  with  deep  waters,  and  that 


259  THEORY    OF    THE     TIDES. 

one  side  of  it  is  stationary  towards  the  moon,  for  any  con- 
siderable length  of  time,  we  might  well  suppose  the  earth 
would  become  elongated  in  the  direction  towards,  and  from  the 
moon,  to  the  extent  of  the  moon's  polarizing  influence. 

But  if  the  whole  globe  were  formed  of  water,  or  if  it  were  cov- 
ered with  deep  waters,  and  were  to  revolve  upon  its  axis,  as  it  now 
does,  it  may  be  hard  to  conceive  how  a  tide  could  well  be  formed  ; 
for  notwithstanding  the  earth  would  be  polarized  in  a  direction  to 
and  from  the  moon,  yet,  as  those  poles  must  travel  on  the  sur- 
face of  the  earth,  at  the  rate  of  about  one  thousand  miles  an 
hour,  in  consequence  of  the  diurnal  motion  of  the  earth,  there 
would  be  no  time  for  the  accumulation  of  the  waters  at  those 
moving  poles  ;  and  hence  the  waters  must  remain  at  rest,  without 
rise,  or  without  reciprocation  of  rise  and  fall.  Nor  can  I  conceive 
how  the  most  imaginative  or  fanciful  theorist,  could,  under  those 
circumstances,  assign  the  place  where  the  waters  should  rise,  or 
the  reciprocation  occur. 

But  the  surface  of  the  earth  being  divided  into  oceans  and 
continents,  and  the  matter  upon  the  surface  of  the  earth  having 
a  constant  tendency  (caused  by  the  moon)  to  accumulate  at  the 
poles  ;  hence,  when  the  moon  is  passing  over  a  broad  continent 
and  vast  solid  parts  of  the  surface  are  under  the  influence  of  this 
polarization  for  a  long  time,  or  for  several  hours,  the  continent 
being  itself  fixed  and  immovable  upon  the  surface  of  the  earth, 
serves  to  produce  an  effect  similar  to  that  which  would  be  pro- 
duced, if  the  whole  surface  were  covered  with  deep  waters,  and  if 
one  side  of  the  earth  should  be  made  to  remain  stationary  un- 
der the  moon  for  hours  together. 

That  is,  the  waters  of  the  surface  of  the  oceans  adjoining  the 
continent  over  which  the  moon  should  be  passing,  (and  like  phe- 
nomena would  occur  on  the  side  of  the  earth  opposite  the  moon,) 
would  flow  or  tend  towards  such  polarized  continent,  and  pile 
upon  its  shores,  ascend  its  rivers,  &c. 

And  hence,  the  flowing  of  the  tides  is  always  towards  con- 
tinents or  large  bodies  of  land,  caused  by  the  fixedness  of 
the  land,  and  its  polarization  by  the  moon ;  and  the  height 
of  the  tide,  or  the  piling  of  the  waters  upon  the  shores,  will 
be  mostly  governed  by  the  magnitude  or  breadth  of  the  continent, 
and  the  situation  of  the  moon  in  respect  to  the  continent. 

So  also  the  flowing  of  the  tide  may  be  east,  west,  south  0*' 
north,  or  in  any  direction,  as  the  body  of  land  shall  dictate ;  and 
as  the  flowing'  of  the  tide  is  towards  the  land,  its  ebb  is  necessa- 
rily from  the  land  ;  and  hence  the  rising  and  falling  of  what  is 
termed  the  great  tidal  ivave,  by  reciprocation,  seems  to  savor 
more  of  fancy  than  of  possibility  or  reality. 


THEORY    OF    THE     TIDES.  257 

Upon  this  hypothesis,  the  waters  would  rise  upon  the  shores  of 
islands  near  continents,  the  same  as  upon  the  snores  of  the  con- 
tinents ;  but  in  the  middle  of  broad  oceans,  or  far  from  large 
bodies  of  land,  the  tides  would  be  small;  so  also  on  the  shores 
of  continents,  where  the  continent  was  narrow,  &c. 

While  this  hypothesis  accounts  for  the  tides  being  greater  on 
the  shores  of  continents  than  in  the  broad  oceans,  it  also  accounts 
for  the  reason  of  the  tides  not  being  so  long  behind  the  meridian 
of  the  moon,  in  the  midst  of  broad  oceans,  as  upon  the  shores 
of  continents ;  as  well  as  why  it  is  behind  the  moon's  meridian 
at  all ;  without  resorting  for  a  reason,  to  the  tardiness  of  the 
water  in  obeying  the  power  of  attraction,  arising  from  its  inertia, 
for  the  water  will  continue  to  pile  upon  the  shores  of  a  conti- 
nent, so  long  as  the  continent,  from  polarization,  possesses  an 
attracting  influence  sufficient  to  draw  the  waters  towards  it ;  and 
it  will  not  be  denied,  that  time  is  requisite  for  the  flowing  of 
water,  whether  it  be  in  a  river  or  upon  the  surface  of  the  ocean. 

Should  it  be  objected,  that  the  time  of  high  water  on  the  west 
side  of  a  continent,  should  not  be  so  long  behind  the  meridian 
of  the  moon  as  on  the  east  side,  I  will  only  remark,  that  the 
whole  cause  of  the  tide  arises  from  the  influence  which  polar- 
ized terra  firma  has  upon  that  substance,  (namely  water.)  which 
is  always  at  liberty  to  flow  towards  the  fixed  earth,  as  the  moon, 
from  its  apparent  rapid  motion,  cannot  of  itself  move  the  waters 
at  all.  And  each  side  of  the  continent  is  under  the  influence 
of  the  moon  for  an  equal  length  of  time  ;  and  most  clearly,  such 
objection  would  apply  with  equal,  if  not  greater  force  to  any 
other  theory  extant 

When  the  moon  is  in  conjunction  or  in  opposition,  namely,  in 
syzygies,  the  tides  are  highest ;  and  when  in  quadratures,  the 
tides  are  lowest,  or  very  nearly  at  those  times,  the  altitude  of 
the  tides  at  quadratures  to  those  at  syzygies  being  proportioned 
to  each  about  as  1  to  1.3756,  although  it  is  said  the  ratio  will 
vary,  from  many  circumstances.  The  tides  also  increase  in  alti- 
tude from  quadratures  to  syzygies,  and  diminish  from  syzygies  to 
quadratures  ;  which  increase  has  been  attributed  to  the  attraction 
of  the  sun  conspiring  with  the  moon  at  those  times,  and  that 
the  decrease  is  occasioned  by  the  attraction  of  the  sun,  conspir- 
ing against  the  moon,  or  not  in  the  line  of  direction  with  the 
moon ;  and  hence,  that  the  sun  is  gradually  abstracting,  or 
separating  its  own  tides  from  those  of  the  moon,  as  the  moon 
approaches  its  quadratures  ;  at  which  times  the  respective  tides 
of  the  sun  and  moon  are  in  quadratures,  which  must  necessarily 
throw  the  earth  into  the  shape  of  a  double  oblate  spheroid,  the 
33 


258  THEORY    OF    THE     TIDES. 

altitude  of  the  moon's  tides  being  to  those  of  the  sun's  as  2.662 
to  1,  or  as  5.412  feet  to  2.033  feet. 

That  those  four  tides  may  exist  when  the  moon  is  in  quadra- 
tures, (according  to  the  Newtonian  theory,)  both  the  sun  and 
moon  must  draw  away  the  solid  parts  of  the  earth  from  the  mass 
of  waters  on  the  side  of  the  earth  opposite  the  given  luminary; 
and  hence, .the  motion  of  the  solid  parts  of  the  earth  could  not 
be  in  a  direct  line  toward  either,  (arid  whether  an  equation 
has  ever  been  resorted  to  in  consequence  of  those  directing 
forces,  I  know  not.) 

So  when  the  sun  and  moon  are  in  conjunction,  their  united 
influence  must  draw  the  solid  parts  of  the  earth  directly  towards 
the  sun  and  moon,  leaving  the  mass  of  waters  on  the  opposite 
side. 

But  when  the  sun  and  moon  are  in  opposition,  it  may  be 
hard  to  determine  which  way  the  solid  parts  of  the  earth  will 
be  drawn,  as  the  tides  would  be  equal  under  each  luminary. 

This  then,  must  be  a  very  perplexed  question  ;  nor  do  I  think 
it  in  anywise  solved,  or  the  difficulty  obviated,  by  the  more 
modern  and  supposed  ingenious  theory  of  La  Place. 

For  according  to  some  authors,  the  sun  exerts  about  twice  the 
influence  upon  the  earth,  taking  both  its  external  and  internal 
parts,  that  the  moon  does  ;  "  but  from  its  very  great  distance" 
it  does  not  exert  so  much  upon  the  surface.  But  in  this  case,  it 
would  seem  that  it  might  exert  its  whole  influence  fairly,  as  it  is 
the  internal  and  solid  parts  of  the  earth  that  are  to  be  drawn 
away  from  the  opposite  waters. 

It  is  alleged  by  some,  that  as  the  moon  subtends  a  greater 
angle  than  the  sun  does,  when  viewed  from  the  earth,  its  in- 
fluence upon  the  earth  is  greater  than  that  of  the  sun ;  and 
having  concluded,  from  the  amount  of  tide  raised  by  each, 
(having  no  other  method  whereby  to  ascertain  the  fact,)  that  the 
density  of  the  moon  to  that  of  the  sun  is  about  as  2.5  to  1,  and 
hence,  that  the  difference  between  their  attractive  influences  upon 
the  earth  where  their  respective  subtended  angles  and  densities 
are  compounded,  is  about  as  2.662  to  1,  by  which  it  would 
seem  that  the  moon  has  quite  the  advantage  over  the  sun  in  its 
attractive  influence  upon  the  earth. 

But  to  me  all  the  theories  extant  in  respect  to  the  tides  seem 
to  be  fraught  with  inexplicable  dilemmas.  I  will  therefore  leave 
them,  and  will  return  and  finish  my  own. 

If  then  the  sun  exerts  an  influence  on  the  moon  wholly  re- 
gardless of.  the  fixed  pole  or  poles  of  the  moon,  and  similar  to 
that  exerted  by  the  moon  upon  the  earth,  causing  two  moving 
or  revolving  poles  of  the  moon,  always  directed  to  and  from  the 


THEORY    OF    THE    TIDES.  259 

sun,  performing  their  revolution  in  the  same  time  that  the  moon 
is  revolving  about  the  earth,  or  in  revolving  upon  its  axis  in 
respect  to  the  sun ;  then,  when  the  sun  and  moon  are  either  in 
conjunction,  or  in  opposition,  one  of  the  movable  poles  of  the 
moon  will  be  extended  directly  towards  the  earth,  in  which  case 
its  full  influence  will  be  exerted  in  connection  with  the  fixed 
pole  of  the  moon,  upon  the  earth,  in  increasing  the  influence  of 
its  movable  poles ;  and  hence,  an  increase  of  the  tides.  But 
as  the  moon  proceeds  towards  quadratures,  the  influence  of  the 
movable  pole  of  the  moon  upon  the  earth  becomes  gradually 
abstracted,  as  the  line  of  its  direction  varies  more  and  more  from 
the  earth,  until  quadratures ;  after  which,  the  opposite  movable 
pole  of  the  moon  commences  and  increases  in  influence  until 
syzygies ;  and  so  on,  alternating  from  syzygies  to  quadratures, 
and  from  quadratures  to  syzygies.  And  hence,  no  portion 
of  the  tides  is  caused  directly  by  the  sun,  nor  in  fact,  by  the 
moon ;  their  influence  being  only  exerted  in  polarizing  the  earth ; 
which  polarization  can  only  have  an  effect  in  raising  the  tides 
when  situated  in  a  continent,  or  terra  firma,  against  which  the 
flowing  waters  may  pile. 

This  theory  then,  relieves  from  all  disturbing  forces  incident 
to  other  theories  of  the  tides;  from  the  hypothesis  that  the  earth 
is  constantly  drawn  from  a  true  curve,  and  hence  performing  its 
revolution  in  wholly  an  inexplicable  orbit ;  performing  "  cycles 
and  epicycles  "  in  consequence  of  the  attraction  of  the  moon ; 
and  even  from  being  disturbed  in  its  regular  course  by  any 
disturbing1  force  of  the  sun. 


CHAPTER   VII. 


An  Investigation  of  the    Theory  of  the  Progressive  Motion  of 

Light. 

PERHAPS  neither  the  conjecture  of  Cassini  the  elder,  (that  acute 
observer  in  astronomy,)  that  light  was  progressive  in  time,  from 
observations  by  him  made  on  Jupiter's  first  satellite,  (which  con- 
jecture, however,  he  shortly  abandoned,)  nor  the  supposition 
of  Roemer  from  a  like  cause,  that  light  is  progressive  in  time, 
would  ever  have  been  adopted  into  philosophy  as  an  infallible 
truth,  and  been  the  cause  of  so  much  time  and  trouble  in  en- 
deavoring to  equate  the  time  in  which  light  is  conceived  to  be 
passing,  had  not  Dr.  Bradley,  (astronomer  royal  to  George  II.,) 
some  few  years  after  the  supposed  discovery  by  Roemer,  come 
to  the  conclusion  of  a  progressive  motion  of  light,  from  being 
unable  otherwise  to  account  for  an  apparent  motion  in  the  fixed 
stars,  some  of  which,  he  had  been  very  critically  and  carefully 
examining  with  a  view  to  ascertain  whether  they  had  an  appa- 
rent parallax  or  motion.  From  these  observations  he  came 
to  the  conclusion,  (which  the  world  have  ever  since  assented  to,) 
that  light  had  a  progressive  motion  in  time ;  and  that  its  velocity 
could  be  determined  by  his  method,  with  far  greater  accuracy 
and  certainty  than  could  be  hoped  for  or  expected  from  obser- 
vations made  upon  Jupiter's  satellites.  Since  this  time,  in 
adapting  equations,  (even  in  respect  to  Jupiter's  satellites,)  re- 
course has  been  had  to  Dr.  Bradley's  hypothesis  in  making 
those  equations,  so  far  as  the  motion  of  light  is  to  be  considered. 

But  there  are  many  reasons  or  causes  why  Jupiter's  satellites 
furnish  no  conclusive  evidence  of  the  progressive  motion  of 
light,  which  naturally  fall  out  from  those  astronomers  who  have 
most  observed  the  satellites  with  a  view  to  furnish  correct  tables 
of  their  eclipses,  for  the  purpose  of  aiding  the  science  of  as- 
tronomy, and  for  the  better  ascertaining  or  finding  of  longitude. 
And  it  is  even  possible  that  from  the  equations  of  light  which 
they  have  been  led  to  adopt,  they  have  also  been  led,  for  the 
purpose  of  forming  correct  tables,  to  adopt  other  and  even  em- 


PROGRESSIVE    MOTION   OF    LIGHT.  261 

pyric  equations,  which  otherwise  would  have  been  dispensed 
with.  Some  of  the  causes  why  Jupiter's  satellites  would  not 
be  a  safe  criterion  whereby  to  calculate  the  progressive  motion 
of  light,  I  will  here  recount. 

1.  The  various  modes  or  methods  of  ascertaining  a  synodic 
revolution  of  the  satellite,  as  by  an  eclipse ;  the  passage  of  the 
satellite  over  Jupiter's  disk,  &c.,  which  may  lead  to  more  or  less 
error. 

2.  In  an  eclipse,  the  satellite  does  not  always  pass  through 
the   centre  of  Jupiter's  shadow;  hence  from  this   cause  some 
eclipses  are  longer  than  others. 

3.  Jupiter  is  a  spheroid,  and  its  shadow  an  ellipse ;  and  the 
position  of  its  shadow  is  not  always  the  same  in  respect  to  the 
satellite. 

4.  The  immersions  are  observed  before  Jupiter's  opposition, 
and  the  emersions  after ;  and  those  must  be  compared  with  each 
other  at  distant  times  in  order  to  get  their  mean,  or  the  true  con- 
junction of  the  satellite. 

5.  Some  of  the  satellites  are  said  to  be  longer  at  some  times 
than  at  others  in  passing  through  the  centre  of  Jupiter's  shadow, 
on  account  of  the  eccentricity  of  their  orbits ;  and  that  the  dura- 
tion of  eclipses  are  unequal. 

6.  It  is  said  that  from  the  unequal  motion  of  Jupiter,  the  true 
conjunctions  of  the  satellites  compared  with  the  mean  may  vary 
by  twice  the  greatest  equation  of  Jupiter's  orbit,  which  amounts 
to  near  forty  minutes  in  the  first  satellite,  and  more  than  an  hour 
in  the  second.     And  on  this  equation  a  mean  synodic  revolution 
as  well   as   the   eclipses   must    depend.     And    even  this  equa- 
tion must  be  variable,  for  the  reason  that  the  eccentricity  and 
greatest  equation  of  Jupiter's  orbit  is  subject  to  change. 

7.  After  making  the  equations  for  the  supposed  motion  of 
light,  a  number  of  empyric  equations  are  necessary  in  order  to 
make  the  tables  and  the  theory  agree  with  observation. 

8.  Maraldi,  whose  tables  have  been  esteemed,  observed  that 
even  Cassini's  tables  erred,  as  to  the  time  of  the  fourth  satellite, 
near  two  hours,  for  which  error  several  astronomers  have  at- 
tempted to  assign  a  cause,   as  the  eccentricity  of  its  orbit,  the 
situation  of  Jupiter  in  its  orbit,  &c. 

9.  Error  of  time  in  calculating  the  entry  of  the  satellite  into 
the  shadow,  or  upon  his  disk;  and  that  the  calculations  deduced 
therefrom  are  different  by  different  observers,  and  said  not  to  be 
depended  upon. 

10.  The  first  satellite  is  said  by  astronomers  to  be  the  best  for 
finding  the  longitude ;   the  tables  of  that  being  more  correct, 
on  account  of    its   vanishing   and    appearing   more   instantly, 
in  an  eclipse,  than  the  others,  and  therefore  being  more  certain. 


262  ON    THE    THEORY    OF    THE 

11.  Astronomers  say  it  is  better  to  compare  eclipses  observed 
under  different  meridians,  than  to  depend  on  the  tables,  on  ac- 
count of  their  incorrectness ;  and  then,  that  the  observer  should 
have  the  same  kind  of  telescope. 

12.  Astronomers  say,  the  time  when  a  satellite  becomes  vis- 
ible or  invisible,  depends  upon  many  circumstances ;    as  the 
telescope  —  the  proximity  of  the  satellite  to  Jupiter  —  the  altitude 
above  the  horizon  —  the  state  of  the  weather,  &c. 

13.  Dr.  Bradley  found  from  observations  made  in  Cassini's 
time,  that  the  eclipses  of  Jupiter's  first  satellite  lasted  six  minutes 
longer  at  some  times  than  at  others,  which  indicates  that  its 
motion  was  not  uniform  ;  but  Bailie  and  some  others,  concluded 
it  arose  from  the  disturbing  force  of  some  of  the  other  satellites. 

14.  Authorities  say,  that  the  satellite  will  disappear  later  and 
appear  earlier,  the  better  the  telescope  is.      And   some   have 
observed,  that  the  disappearance  of  the  satellite  depended   on 
the   distance   of  Jupiter   from   the   earth  and   sun ;  and  others 
have  observed,  that  different  states  of  the   atmosphere,  different 
altitudes,  and  their  distance  from  Jupiter,  have  a  like  influence. 

15.  The  authorities  also  say,  that  the  time  when  the  satellite 
becomes  invisible  at  an  immersion,  or  visible  at  an  emersion, 
depends  upon  the  quantity  of  light,  which  the  telescope  receives, 
and  its  magnifying  power,  as  well  as  on  other  causes,  —  as  upon 
the   altitude   of   Jupiter    above    the  horizon,   the    twilight,  the 
clearness  of  the  air,  the   proximity  of    Jupiter  to  the  moon,  and 
the  eye  of  the  observer ;    all  combining  to  affect  the  time  at 
which  the  satellite  becomes  visible  or  invisible. 

It  is  highly  probable  that  Cassini  doubted  the  correctness  of 
his  own  observations,  so  far  at  least  as  to  cast  away  the  idea  of 
the  progressive  motion  of  light,  which  Roemer  picked  up  and 
adopted. 

Cassini  only  suspected  the  motion  of  light,  from  observing 
that  the  immersions  of  Jupiter's  first  satellite  took  place  sooner 
and  sooner  in  respect  to  the  computed  time,  as  they  are  observed 
from  the  conjunction  of  Jupiter  to  his  opposition ;  and  that  the 
emersions  took  place  later  and  later,  as  they  are  observed  from 
Jupiter's  opposition  to  his  conjunctipn. 

Now  good  philosophy  would  dictate,  as  it  did  to  Cassini, 
that  it  would  not  be  proper  to  ingraft  a  hypothesis  or  theory  into 
the  code  of  philosophy,  upon  so  slight  or  doubtful  evidence  as 
the  small  apparent  difference  between  the  observed  and  computed 
time  of  a  revolution  of  Jupiter's  first  satellite,  lest  we  adopt  false 
premises,  and  deduce  wrong  conclusions,  which  will  only  serve 
to  bewilder  and  perplex  us,  in  accounting  for  phenomena,  and 
in  making  those  necessary  and  rational  deductions  and  conclu- 


PROGRESSIVE    MOTION    OF    LIGHT.  263 

sions,  so  necessary  in  support  of  a  true  theory ;  while  hasty 
and  absurd  conclusions  or  deductions  often  serve  only  to  lead 
us  into  a  thousand  empyric  adaptations  and  adjustments,  in 
making  out  and  establishing  our  theories,  from  having  in  the 
outset  overlooked  some  important  point  in  the  logical  analogies 
of  nature,  which  eventually  lead  on  to  false  deductions. 

From  the  foregoing,  and  from  many  other  causes  and  diffi- 
culties that  have  existed  and  still  exist,  in  the  forming  of  correct 
tables  of  the  eclipses  of  Jupiter's  satellites,  it  may  well  be 
doubted,  (as  I  have  already  remarked,)  whether  the  theory  of  a 
progressive  motion  of  light,  would  have  been  adopted  as  an 
established  truth  in  philosophy,  from  a  renewal  by  Roemer  of 
the  momentary  and  cast  off  hypothesis  of  Cassini,  had  not  Dr. 
Bradley  soon  after  seized  upon  the  hypothesis,  to  aid  him  in 
accounting  for  an  apparent  motion  of  the  fixed  stars;  which 
was  supposed  not  only  to  confirm  the  theory  of  Roemer,  but  to 
a  degree  of  exactness  and  certainty  that  could  never  be  hoped 
for  or  expected  from  Jupiter's  satellites. 

But  fully  consenting  to  the  very  careful  and  critical  observa- 
tions of  Dr.  Bradley,  upon  the  fixed  stars,  and  their  apparent 
motions,  as  by  him  observed,  should  his  theoretic  conclusions 
deduced  therefrom  prove  unphilosophical  and  absurd,  on  further 
investigation,  it  may  yet  be  hoped  that  we  shall  again  observe 
the  sun  in  its  true  place  ;  that  we  shall  see  it  where  it  is,  and 
not  where  it  is  not ;  and  that  astronomers  will  be  relieved  from 
a  thousand  empyric  equations,  adopted  for  the  purpose  of  making 
error  as  beautiful  as  truth. 

I  now  propose  to  examine  the  evidence  furnished  by  Dr. 
Bradley,  in  support  of  the  theory  of  the  progressive  motion  of 
light  in  time,  as  deduced  from  his  observations.  And  in  doing 
so,  I  shall  avail  myself  of  the  common  terms  and  phrases 
usually  applied  to  light,  considered  both  as  a  substance,  and 
progressive  in  time,  as  the  rays  of  light,  the  particles  of  light, 
the  motion  of  light,  the  emission  of  light,  &c. ;  believing,  however, 
that  a  better  vocabulary  of  terms,  as  applied  to  light,  might  be 
adopted,  in  which  specific  terms  would  not  convey  to  the  mind 
a  whole  body  of  false  philosophy. 

I  will  not  say  that  I  am  intending  to  prove  a  negative,  —  to 
prove  that  light  is  not  progressive  in  time,  or  that  time  is  not 
necessary  to  the  perception  of  a  distant  luminous  body ;  but 
rather,  that  sufficient  facts  or  data  have  not  as  yet  been  ascer- 
tained, to  prove  that  light  is  progressive  in  time.  And  should 
those  views  prove  correct,  they  will  further  serve  to  show  the 
propriety  of  giving  to  a  hypothesis  a  logical  investigation  previ- 
ous to  applying  to  it  mathematical  and  geometric  adaptation. 


264  ON    THE    THEORY    OF    THE 

and  supposed  demonstration,  lest  paradoxes,  dilemmas  and 
absurdities,  intrude  themselves,  to  the  overturning  of  the  theory, 
if  it  be  not  true  in  fact. 

Dr.  Bradley's  observations  upon  the  fixed  stars,  from  whence 
he  derived  his  proof  as  to  the  progressive  motion  of  light,  were 
made  as  early  as  the  year  1728,  an  account  of  which  he  com- 
municated to  Dr.  Halley,  in  a  letter  containing  the  results  of  his 
observations,  and  the  final  conclusions  he  had  drawn  from  the 
same.  And  that  Dr.  Bradley  may  be  heard  in  respect  to  his 
observations  upon  the  fixed  stars,  together  with  his  deductions 
drawn  therefrom,  I  will  here  give  such  parts  of  his  letter  as  directly 
refer  to  or  serve  to  show  his  manner  of  observation,  together 
with  his  conjectures  and  conclusions  ;  not  quoting  those  portions 
of  his  letter  which  only  serve  to  show  the  method  of  fixing  up 
his  instrument,  &c.  He  says  : 

"  Mr.  Molyneux's  apparatus  (a  zenith  sutor)  was  completed  and 
fitted  for  observing,  about  the  end  of  November,  1725,  and  on 
the  third  day  of  December  following,  the  bright  star  in  the  head 
of  Draco,  (marked  y  by  Bayer,  called  Draconis,)  was  for  the  first 
time  observed,  as  it  passed  near  the  zenith,  and  its  situation 
carefully  taken  with  the  instrument.  The  like  observations 
were  made  on  the  5th,  llth,  and  12th  of  the  same  month  ;  and 
there  appearing  no  material  difference  in  the  place  of  the  star, 
a  farther  repetition  of  them  at  this  season  of  the  year,  seemed 
needless,  it  being  a  part  of  the  year  wherein  no  sensible  altera- 
tion of  parallax  in  this  star  could  soon  be  expected.  It  was 
chiefly,  therefore,  curiosity  that  tempted  me,  (being  then  at  Kew, 
where  the  instrument  was  fixed,)  to  prepare  for  observing  the 
star  on  December  17th,  when,  having  adjusted  the  instrument  as 
usual,  I  perceived  that  it  passed  a  little  more  southwardly  this 
day,  than  when  it  was  observed  before.  Not  suspecting  any 
other  cause  of  this  appearance,  we  at  first  concluded  that  it  was 
owing  to  the  uncertainty  of  the  observations,  and  that  either  this 
or  the  foregoing  was  not  so  exact  as  we  had  before  supposed  ;  for 
which  reason,  we  purposed  to  repeat  the  observation  again,  in 
order  to  determine  from  whence  this  difference  proceeded ;  and 
upon  doing  it  on  December  20th,  I  found  that  the  star  passed 
still  more  southwardly  than  in  the  former  observations.  This 
sensible  alteration  the  more  surprised  us,  in  that  it  was  the  con- 
trary way  from  what  it  would  have  been,  had  it  proceeded  from 
an  annual  parallax  of  the  star  ;  but  being  now  pretty  well  satis- 
fied that  it  could  not  be  entirely  owing  to  the  want  of  exactness 
in  the  observations,  and  having  no  notion  of  anything  else  that 
could  cause  such  an  apparent  motion  as  this  in  the  star,  we 
began  to  think  that  some  change  in  the  materials,  &c.,  of  the 


PROGRESSIVE    MOTION    OF    LIGHT.  265 

instrument  itself  might  have  occasioned  it.  Under  these  appre- 
hensions we  remained  some  time,  but  being  at  length  fully 
convinced,  by  several  trials,  of  the  great  exactness  of  the  instru- 
ment, and  finding  by  the  gradual  increase  of  the  star's  distance 
from  the  pole,  that  there  must  be  some  regular  cause  that 
produced  it,  we  took  care  to  examine  nicely,  at  the  time  of  each 
observation,  how  much  it  was ;  and  about  the  beginning  of 
March,  1726,  the  star  was  found  to  be  20"  more  southwardly 
than  at  the  time  of  the  first  observation.  It  now  indeed  seemed 
to  have  arrived  at  its  utmost  limit  southward,  because  in  sev- 
eral trials  made  about  this  time,  no  sensible  difference  was 
observed  in  its  situation." 

By  the  middle  of  April,  it  appeared  to  be  returning  back 
again  towards  the  north ;  and  about  the  beginning  of  June  it 
passed  at  the  same  distance  from  the  zenith  as  it  had  done  in 
December,  when  it  was  first  observed.  "  From  the  quick  altera- 
tion of  this  star's  decimation  about  this  time,  (it  increasing  a 
second  in  three  days,)  it  was  concluded  that  it  would  now  pro- 
ceed northward,  as  it  before  had  gone  southward  of  its  present 
situation  ;  and  it  happened  as  was  conjectured,  for  the  star  con- 
tinued to  move  northward  till  September  following,  when  it 
again  became  stationary,  being  then  near  20"  more  northwardly 
than  in  June,  and  no  less  than  39"  more  northwardly  than  it  was 
in  March.  From  September  the  star  returned  towards  the  south, 
till  it  arrived,  in  December,  to  the  same  situation  it  was  in  at  that 
time  twelve  months,  allowing  for  the  difference  of  declination 
on  account  of  the  precession  of  the  equinox. 

"  This  was  a  sufficient  proof  that  the  instrument  had  not  been 
the  cause  of  this  apparent  motion  of  the  star,  and  to  find  one 
adequate  to  such  an  effect  seemed  a  difficulty.  A  nutation  of 
the  earth's  axis  was  one  of  the  first  things  that  offered  itself  upon 
this  occasion ;  but  it  was  soon  found  to  be  insufficient ;  for 
though  it  might  have  accounted  for  the  change  of  declination  in 
7  Draco nis,  yet  it  would  not  at  the  same  time  agree  with  the 
phenomena  in  other  stars,  particularly  in  a  small  one  almost 
opposite  in  right  ascension  to  7  Draconis,  at  about  the  same  dis- 
tance from  the  north  pole  of  the  equator;  for  though  this  star 
seemed  to  move  the  same  way  as  a  nutation  of  the  earth's  axis 
would  have  made  it,  yet  it  changed  its  declination  but  about 
half  as  much  as  7  Draconis  in  the  same  time,  (as  appeared  by 
comparing  the  observations  of  both  made  upon  the  same  days 
at  different  seasons  of  the  year ;)  this  plainly  proved  that  the 
apparent  motion  of  the  stars  was  not  occasioned  by  a  real  nu- 
tation, since  if  that  had  been  the  case,  the  alteration  in  both  stars 
would  have  been  nearly  equal. 
34 


266 


ON    THE    THEORY    OF    THE 


"  The  great  regularity  of  the  observations  left  no  room  to 
doubt  but  that  there  was  some  regular  cause  that  produced  this 
unexpected  motion,  which  did  not  depend  on  the  uncertainty 
or  variety  of  the  seasons  of  the  year.  Upon  comparing  the  ob- 
servations with  each  other,  it  was  discovered  that  in  both  the 
fore-mentioned  stars,  the  apparent  difference  of  declination  from 
the  maxima  was  always  nearly  proportional  to  the  versed  sine 
of  the  sun's  distance  from  the  equinoctial  points.  This  was  an 
inducement  to  think  that  the  cause,  whatever  it  was,  had  some 
relation  to  the  sun's  situation  with  respect  ta  those  points.  But 
not  being  able  to  frame  any  hypothesis  at  that  time  sufficient  to 
solve  all  the  phenomena,  and  being  very  desirous  to  search  a 
little  farther  into  this  matter,  I  began  to  think  of  erecting  an 
instrument  for  myself  at  Wanstead,"  &c. 

After  giving  the  preparation  of  his  instrument,  he  says,  "  my 
instrument  being  fixed,  I  immediately  began  to  observe  such 
stars  as  I  judged  most  proper  to  give  me  light  into  the  cause  of 
the  motion  already  mentioned.  There  was  variety  enough  of 
small  ones,  and  not  less  than  twelve  that  I  could  observe  through 
all  the  seasons  of  the  year,  they  being  bright  enough  to  be  seen 
in  the  daytime  when  nearest  the  sun.  I  had  not  been  long 
observing  before  I  perceived  that  the  notion  we  had  before  en- 
tertained of  the  stars'  being  farthest  north  and  south  when  the 
sun  was  about  the  equinoxes,  was  only  true  of  those  that  were 
near  the  solstitial  coloure ;  and  after  I  had  continued  my  obser- 
vations a  few  months,  I  discovered  what  1  then  apprehended  to 
be  a  general  law,  observed  by  all  the  stars,  viz.,  that  each  of 
them  became  stationary  or  was  farthest  north  or  south  when 
they  passed  over  my  zenith  at  six  o'clock  either  in  the  morning 
or  evening.  I  perceived  likewise  that  whatever  situation  the  stars 
were  in  with  respect  to  the  cardinal  points  of  the  ecliptic,  the 
apparent  motion  of  every  one- tended  the  same  way  when  they 
passed  my  instrument  about  the  same  hour  of  the  day  or  night ; 
for  they  all  moved  southward  while  they  passed  in  the  day,  and 
northward  in  the  night;  so  that  each  was  farthest  north  when  it 
came  about  six  o'clock  in  the  evening,  and  farthest  south  when 
it  came  about  six  in  the  morning. 

"  Though  I  have  since  discovered  that  the  maxima  in  most  of 
these  stars  do  not  happen  exactly  when  they  come  to  my  instru- 
ment at  those  hours,  yet  not  being  able  at  that  time  to  prove  the 
contrary,  and  supposing  that  they  did,  I  endeavored  to  find  out 
what  proportion  the  greatest  alterations  of  declination  in  differ- 
ent stars  bore  to  each  other;  it  being  very  evident  that  they 
did  not  all  change  their  declinations  equally. 

"  I  have  before  taken  notice  that  it  appeared  from  Mr.  Moly- 


PROGRESSIVE    MOTION    OP    LIGHT.  267 

neux's  observations  that  y  Draconis  altered  its  declination  about 
twice  as  much  as  the  fore-mentioned  small  star  almost  opposite 
to  it ;  but  examining  the  matter  more  particularly,  I  found  that 
the  greatest  alteration  of  declination  in  these  stars  was  as  the 
sine  of  the  latitude  of  each  respectively.  This  made  me  sus- 
pect that  there  might  be  the  like  proportion  between  the  maxima 
of  other  stars;  but  finding  the  observations  of  some  of  them 
would  not  perfectly  correspond  with  such  an  hypothesis,  and 
not  knowing  whether  the  small  difference  I  met  with  might  not 
be  owing  to  the  uncertainty  and  error  of  the  observations,  I 
deferred  the  farther  examination  into  the  truth  of  this  hypothesis 
till  I  should  be  furnished  with  a  series  of  observations  made  in 
all  parts  of  the  year;  which  might  enable  me  not  only  to  deter- 
mine what  errors  the  observations  are  liable  to,  or  how  far  they 
may  safely  be  depended  upon ;  but  to  judge  whether  there  had 
been  any  sensible  change  in  the  parts  of  the  instrument  itself. 

"  Upon  these  considerations  I  laid  aside  all  thoughts  at  that 
time  about  the  cause  of  the  fore-mentioned  phenomena,  hoping 
that  I  should  the  easier  discover  it  when  I  was  better  provided 
with  proper  means  to  determine  more  precisely  what  they  were. 

"  When  the  year  was  completed  I  began  to  examine  and 
compare  my  observations,  and  having  pretty  well  satisfied  my- 
self as  to  the  general  laws  of  the  phenomena,  I  then  endeavored 
to  find  out  the  cause  of  them.  I  was  already  convinced  that 
the  apparent  motion  of  the  stars  was  not  owing  to  a  nutation 
of  the  earth's  axis.  The  next  thing  that  offered  itself,  was  an 
alteration  in  the  direction  of  the  plumb  line  with  which  the  in- 
strument was  constantly  rectified;  but  this  upon  trial  proved 
insufficient.  Then  I  considered  what  refraction  might  do, 
but  here  also  nothing  satisfactory  occurred.  At  last  I  conject- 
ured that  all  the  phenomena  hitherto  mentioned,  proceeded  from 
the  progressive  motion  of  light  and  the  earth's  annual  motion  in 
its  orbit.  For  I  perceived  if  light  was  propagated  in  time,  the 
apparent  place  of  a  fixed  object  would  not  be  the  same  when 
the  eye  is  at  rest,  as  when  it  is  moving  in  any  other  direction 
than  that  of  the  line  passing  through  the  eye  and  object ;  and 
that  when  the  eye  is  moving  in  different  directions,  the  apparent 
place  of  the  object  would  be  different." 

This  is  "  Dr.  Bradley's  account  of  this  very  important  discov- 
ery," and  many  pages  are  filled  with  supposed  geometrical  and 
mathematical  explanations  or  demonstrations  of  the  truth  of  his 
hypothesis,  and  its  adaptation  to  the  equation  of  light,  &c.,  to 
which  explanations  and  adaptations  I  must  refer  all  who  may 
be  curious  to  know  the  vast  amount  of  labor  and  investigation 
that  has  been  bestowed  upon  this  hypothesis. 


268  ON    THE    THEORY    OF    THE 

I  have  thought  proper  to  bring  forward  Dr.  Bradley's  own  ac- 
count of  his  observations,  and  his  theory  or  hypothesis  deduced 
therefrom,  that  they  may  be  readily  compared  with  my  attempt- 
ed refutation  of  his  theory,  and  with  that  which  I  may  suggest 
in  its  stead. 

We  find  also  in  the  works  on  astronomy  this  further  reference 
to  Dr.  Bradley's  observations  and  deductions  upon  the  same 
subject,  viz. 

Dr.  Bradley  further  observes  that  in  seventy  observations 
made  in  a  year  on  7  Draconis,  there  was  but  one  (and  that  is 
noted  very  dubious  on  account  of  clouds)  which  differed  more 
than  2"  from  the  theory,  and  that  did  not  differ  3".  And  in  about 
fifty  observations  made  in  a  year  on  n  Ursce  majoris,  he  did  not 
find  a  difference  of  2",  except  in  one  marked  doubtful  on  ac- 
count of  the  undulation  of  the  air,  &c.  and  that  did  not  differ 
3".  And  this  agreement  between  the  theory  and  observation 
leaves  no  room  to  doubt  but  that  the  cause  is  rightly  assigned. 
And,  if  this  be  the  case,  the  annual  parallax  of  the  fixed  stars 
must  be  extremely  small.  "  I  believe,"  says  the  Dr.  "  that  I  may 
venture  to  say  that  in  either  of  the  above  mentioned  stars,  it  does 
not  amount  to  2".  I  believe  if  it  were  1",  I  should  have  per- 
ceived it  in  the  great  number  of  observations  that  I  made, 
especially  on  /  Draconis ;  which  agreeing  with  the  theory  (with- 
out allowing  anything  for  parallax)  nearly  as  well  in  conjunc- 
tion with,  as  in  opposition  to  this  star,  it  seems  very  probable 
that  the  parallax  is  not  so  great  as  one  single  second ;  and  conse- 
quently that  it  is  above  four  hundred  thousand  times  further  from 
us  than  the  sun."  The  observations  of  Mr.  Flamstead  of  the 
different  distances  of  the  pole  star  from  the  pole  at  different  times 
of  the  year,  and  which  was  looked  upon  as  a  proof  of  its  an- 
nual parallax,  was  undoubtedly  owing  to  this  cause.  For  he 
concluded  that  the  star  was  35",  40"  or  45"  nearer  the  pole  in 
December  than  in  May  or  July;  and  according  to  this  hypothe- 
sis, it  ought  to  appear  40"  nearer  in  December  than  in  June. 
This  agreement  is  greater  than  could  have  been  expected  from 
observations  made  with  his  instrument." 

Hence  Dr.  Bradley  deduced  the  following  conclusions  :  1. 
"  That  the  light  of  all  the  fixed  stars  arrives  at  the  earth  with 
equal  velocities ;  for  the  major  axis  of  the  ellipse  is  the  same  in  all 
the  stars,  that  is  40",  according  to  his  last  determination.  2.  That 
unless  their  distances  from  us  are  all  equal,  which  is  very  im- 
probable, their  lights  are  propagated  uniformly  to  all  distances 
from  them.  3.  That  light  moves  from  the  sun  to  the  earth  in 
8'  7"  .5,  and  its  velocity  is  to  the  velocity  of  the  earth  in  its  orbit 
as  10314  :  1.  4.  That  the  time  thus  determined  can  scarcely  err 


PROGRESSIVE    MOTION    OF    LIGIJT.  269 

from  the  truth  by  above  5"  or  10"  at  most,  which  is  such  a  de- 
gree of  exactness  as  can  never  be  expected  from  the  eclipses  of 
Jupiter* s  satellites.  5.  That  as  this  velocity  of  star  light  comes 
out  about  a  mean  of  the  several  velocities  found  from  the 
eclipses  of  Jupiter's  satellites,  we  may  reasonably  conclude  that 
the  velocity  of  these  reflected  lights  are  equal  to  the  velocity  of 
direct  light.  6.  And  as  it  is  highly  probable  that  the  velocity  of 
the  sun's  emitted  light  is  equal  to  that  of  star  light,  it  follows 
that  its  velocity  is  not  altered  by  reflection  into  the  same  medium." 
I  have  thought  proper  to  insert  the  foregoing  reference  to  Dr. 
Bradley's  observations  and  theory,  as  furnishing  (after  most  am- 
ple applications  to  practical  utility  as  is  conceived)  a  string  of 
corollaries  extending  to  such  things  as  were  conceived  to  fall 
within  analogies. 

Dr.  Bradley  in  rendering  a  hypothesis  to  the  phenomena 
which  were  the  result  of  his  observations,  has  made  this  assump- 
tion. "  He  perceived,  that  if  light  was  propagated  in  time,  the 
apparent  place  of  a  fixed  object  would  not  be  the  same  when  the 
eye  is  at  rest,  as  when  it  is  moving  in  any  other  direction  than 
that  of  the  line  passing  through  the  eye  and  object ;  and  that 
when  the  eye  is  moving  in  different  directions,  the  apparent 
place  of  the  object  would  be  different." 

Now  it  appears  to  me  this  is  begging  the  question  ;  for  not- 
withstanding the  Dr.  has  carefully  inserted  the  word  if  as  a  salvo, 
yet  the  world  have  assumed  the  premises  as  being  true,  and  have 
reasoned  more  critically  on  the  subject  than  I  am  able  to  com- 
prehend as  being  either  sound  or  true,  notwithstanding  their  ex- 
planation of  the  laws  of  motion  upon  the  principles  of  a  par- 
allelogram or  resolution  of  forces,  and  their  finally  assumed 
proportions  between  the  velocity  of  light,  and  that  of  the  earth 
in  its  orbit,  as  sometimes  attempted  to  be  explained  by  con- 
ceiving a  very  long  telescope  or  tube  of  sufficient  calibre  only 
to  permit  a  single  ray  or  particle  of  light  to  flow  down  in  the 
tube ;  and  so  adjusted  in  respect  to  the  motion  of  light  and  the 
motion  of  the  earth  in  its  orbit,  that  the  ray  or  particle  of  light 
would  flow  freely  down  the  tube  or  telescope  ;  and  that  the  eye 
placed  at  the  lower  end  of  the  tube  would  perceive  the  ray  or 
particle  of  light  giving  the  appearance  of  the  star  or  object  from 
which  the  light  is  emitted,  in  the  line  of  the  tube;  notwithstand- 
ing the  tube  or  telescope  would  not  lie  in  the  line  from  the  eye 
to  the  object ;  but  so  situated  as  to  catch  the  rays  or  particles  of 
light  in  advance  of  such  direct  line  from  the  eye  to  the  object. 
But  this  mode  of  reasoning  appears  to  me  to  overlook  some  of 
the  plainest]  and  best  established  principles  and  facts  in  the 
science  of  optics;  for  although  I  have  long  considered  the 


270  ON    THE    THEORY    OF    THE 

adopted  theories  on  light  to  be  vague  and  unphilosophical,  nev- 
ertheless, the  theory  of  optics,  so  far  as  regards  the  effects  or  op- 
erations of  light,  is  in  many  respects  as  beautifully  and  satisfac- 
torily defined  and  explained  as  that  of  any  other  branch  of 
science  or  philosophy ;  among  its  laws  are  those  which  teach 
us  that  the  eye  discerns  an  object  in  the  precise  direction  in 
which  the  eye  receives  or  obtains  the  light,  whether  the  light 
which  presents  the  object  to  the  eye  be  direct  from  the  object  to 
the  eye,  or  whether  it  may  have  been  reflected  or  refracted ;  but 
when  the  light  that  presents  the  object  to  the  eye  has  been  refract- 
ed, for  the  very  reason  that  the  eye  discerns  the  object  in  the 
exact  direction  in  which  it  receives  the  light  which  presents 
the  object,  the  apparent  place  of  the  object  will  not  be  its  real. 

And  as  well  might  we  expect  to  see  the  sun  at  noon  day  by 
looking  in  any  other  direction  than  towards  it,  (inasmuch  as  its 
rays  are  falling  thick  all  around,)  as  to  perceive  the  star  by  look- 
ing at  those  rays  or  particles  of  light  (as  they  are  called)  which 
are  yet  in  advance  of  a  direct  line  from  the  eye  to  the  star;  and 
the  idea  is  preposterous  that  we  can  see  a  ray  or  particle  of  light 
flowing  down  the  telescope  or  tube,  not  directed  towards  the 
eye,  but  to  a  place  in  advance  of  the  eye,  merely  because  the 
eye  and  ray  or  particle  will  presently  meet  in  the  angle  of  the 
requisite  parallelogram.  And  it  is  not  pretended  even  by  way 
of  explanation  of  Dr.  Bradley's  theory,  that  the  line  of  direction 
of  the  telescope  or  tube  is  in  that  from  the  eye  to  the  star,  but 
as  in  the  case  of  a  composition  of  forces,  an  angle  is  formed  by 
the  line  of  the  eye  and  object  with  the  line  of  the  motion  of  the 
eye,  equal  to  what  is  conceived  to  denote  the  proportional  ve- 
locities of  light,  and  of  the  eye. 

And  to  show  how  the  progressive  motion  of  light  may  affect  the 
place  of  the  star  seen  by  the  naked  eye,  they  have  in  reality,  only 
made  the  eye  a  shorter  tube,  or  telescope ;  and  hence  it  is  said 
that,  "  if  a  ray  of  light  fall  upon  the  eye  in  motion,  its  relative 
motions  in  respect  to  the  eye,  will  be  the  same  as  if  you  were 
to  impress  equal  motions  in  the  same  direction,  upon  each  at 
the  instant  of  contact;  for  equal  motions  in  the  same  direction, 
impressed  upon  two  bodies,  will  not  affect  their  relative  motions, 
and  therefore,  the  effect  of  one  upon  the  other,  will  not  be 
altered."  And  another  parallelogram  of  forces,  or  of  motion,  is 
presented  by  way  of  diagram,  to  exemplify  the  truth  of  the 
position. 

But,  if  philosophers  had  abided  by  all  experience,  and  by 
what  all  sound  philosophy  has  taught,  as  to  the  eternal  fitness  of 
things,  in  adapting  the  mechanism  of  the  eye  to  our  best  conve- 
nience, so  that  when  it  discerns  an  object,  the  crystalline  lens, 
and  other  parts  of  the  eye,  are  so  constructed,  as  to  present  the 


PROGRESSIVE    MOTION   OF    LIGHT.  271 

object  in  a  direct  line  from  the  eye,  unless  the  light  has  been 
reflected,  or  refracted ;  and  which,  of  all  the  laws  of  optics,  is 
most  dear  to  us,  as  presenting  the  object  so  that  we  may  see  it 
where  it  is,  rather  than  where  it  is  not ;  —  they  would,  perhaps,  not 
have  attempted  to  filch  enough  from  the  well  known  and  best 
established  laws  of  optics,  to  make  a  philosophic  truth  of  Dr. 
Bradley's  conjecture  and  final  perception,  that  if  light  was  propa- 
gated in  time,  it  would  solve  all  the  phenomena  arising  from 
his  observations,  by  assuming,  for  the  purpose  of  explanation, 
that  such  is  the  fact;  and  by  those  subtleties  which  may  some- 
times "  make  the  worse  appear  the  better  reason,"  impose  the 
hypothesis  upon  a  credulous  world,  as  being  demonstrated  or 
proved ;  and  therefore  requiring  no  farther  investigation.  But  I 
think,  however  specially,  or  critically,  they  may  have  pleaded  in 
the  case,  that  the  world,  on  a  re-examination  of  the  case,  will 
adjudge  that  they  take  nothing  by  their  plea. 

One  result  of  the  Doctor's  theory  would  be,  that  the  nearer  a 
planet  is  to  the  sun,  the  greater  will  be  the  difference  between 
the  true  and  apparent  place  of  the  sun,  as  observed  from  such 
planet.  This,  however,  would  be  no  direct  or  conclusive  argu- 
ment against  his  theory  or  hypothesis. 

I  will  now  attempt  to  show  some  results,  that  must  necessarily 
follow  or  flow  from  Dr.  Bradley's  hypothesis,  when  tested  by  an 
attempt  at  demonstration.  But,  before  suggesting  a  diagram,  I 
will  quote  from  explanations  of  Dr.  Bradley's  theory,  in  order  to 
justify  myself  in  the  form  or  construction  of  the  diagram  which 
I  have  chosen  to  adopt. 

1.  It  is  assented  to  by  the  authorities,  and  will  be  readily 
assented  to  by  all,  that  for  all  purposes  of  explanation  or  of  demon- 
stration, so  far  as  the  principle  of  the  aberration  of  light  may 
be  concerned,  a  circular  orbit,  with  the  sun  in  the  centre,  is  as 
proper  as   an  eccentric  or  elliptical  orbit ;   the  only  difference 
consisting  in  this  :  that  in  a  circular  orbit,  the  motion  of  the  earth, 
and  of  course,  of  the  eye  of  the  observer,  would  be  uniform ;  while 
in   an  eccentric  orbit,  it  would  be  something  greater  at  the  peri- 
helion than  at  the  aphelion ;  but  even  our  earth's  orbit  is  so 
nearly  a  centric  orbit,  as  to  permit  astronomers  to  consider  it  as 
such,  so  far  as  the  aberration  of  light  is  concerned,  as  well  as 
in  many  other  calculations. 

2.  It  is  said  that,  "  Whilst  the  earth  makes  one  revolution  in 
its  orbit,  the   curve,  parallel  to  the  ecliptic,  described   by  the 
apparent  place  of  a  fixed   star,  is  a  circle," —  to  which  all  will 
assent. 

3.  It  is  said  also,  that  "  in  the  plane  of  the  ecliptic,  the  sine  of 
the  star's  latitude  being  0,  the  orbit  becomes  a  straight  line,"  &c. 


272  ON    THE    THEORY    OF     THE 

That  is,  to  an  observer,  placed  in  the  plane  of  the  ecliptic,  at  a 
great  distance  from  the  line  of  the  earth's  orbit,  if  such  line 
were  a  visible  line,  it  would  also  appear  to  be  a  straight  line. 

4.  It  is  also  said  "  to  find  the  aberration  in  latitude  or 
longitude,  let  A  B  C  D,  be  the  earth's  orbit,  supposed  to  be  a 
circle,  with  the  sun  in  the  centre,"  &c. 

Hence,  in  the  diagram  suggested,  I  shall  conceive  the  star  to 
be  observed,  for  the  purpose  of  getting  its  apparent  motion,  or 
what  is  termed  its  aberration,  to  be  in  the  plane  of  the  ecliptic, 
but  at  such  an  infinite  or  indefinite  distance  from  the  earth,  in 
any  part  of  its  orbit,  as  to  have  no  sensible  parallax,  or  actual 
motion  or  change  of  place,  even  by  the  nicest  observations ;  in 
which  case,  I  think,  the  diagram  will  be  assented  to,  as  wholly 
proper  for  investigating  the  apparent  motion  of  such  star,  caused 
by  the  revolution  of  the  earth  in  its  orbit. 

And  to  avoid  any  inconvenience  or  perplexity  in  considering 
the  different  hours  of  the  day,  when  the  star  observed  comes  to 
the  meridian  where  the  observer  is  situated,  occasioned  by  the 
diurnal  revolution  of  the  earth  on  its  axis,  I  have  conceived  only 
the  eye  of  the  observer,  to  revolve  around  the  sun  in  the  orbit  of 
the  earth,  with  the  same  motion  with  which  the  earth  revolves ; 
in  which  case,  the  eye  of  the  observer  would  at  all  times  see  the 
star ;  and  that  in  such  case,  the  only  apparent  motion  of  such 
star,  would  be  that  of  a  backward  and  forward  motion,  as  in  a 
straight  line,  in  the  plane  of  the  ecliptic,  to  an  amount  of  40"  in 
the  course  of  an  annual  revolution  of  the  eye  of  the  observer. 

I  will  here,  again  recapitulate  some  of  the  phenomena  arising 
from  Dr.  Bradley's  observations,  which  appear  most  essential  to 
be  considered,  with  a  view  to  a  proper  solution. 

1.  The  apparent  motion  of  the  star,  is  always  in  the  same 
direction  in  which  the  earth  would  appear  to  be  moving,  to  an 
observer  situated  at  such  star. 

2.  In  March,  the  star  7  Draconis,  appeared  stationary  for  many 
days,  at  its  utmost  southern  limit. 

3.  In  June,  the   star  had  passed  20"  north  of  its  place  in 
March,  and  was  continuing  northward,  having  then  its  most 
rapid  motion. 

4.  It  continued  to   pass  northward  until  September,  when  it 
was  40"  north  of  its  place  in  March,  and  where  it  again  became 
stationary  for  some  days. 

5.  In  September,  it  commenced  returning  to  the  south,  and  in 
December,  it  was  20"  farther  south  than  it  was   in   September, 
having  again  in  December,  its  most  rapid  motion. 

6.  In  March  following,  it  had  again  arrived  at  its  most  south- 
ern  limit,   and   after    remaining   stationary  for   a  time,   again 
commenced  its  return  to  the  north. 


PROGRESSIVE    MOTION   OF    LIGHT.  273 

Let  us  now  conceive  a  diagram  of  the  orbit  of  the  earth, 
with  its  requisite  appendages,  with  the  star  to  be  observed,  situ- 
ated at  an  indefinitely  great  distance,  but  in  the  plane  of  the  orbit. 

Conceive  a  straight  line  (called  the  zenith  line)  to  be  drawn 
from  the  fixed  star  to  be  observed,  through  the  centre  of  the  orbit, 
intersecting  the  line  of  the  orbit  in  two  points ;  the  point  of  inter- 
section next  the  star  being  denoted  by  A,  and  signifying  June, 
or  the  summer  solstice ;  and  the  opposite  point  of  intersection 
by  C,  and  signifying  December,  or  the  winter  solstice. 

Conceive  another  line  drawn  through  the  centre  of  the  orbit, 
at  right  angles  to  the  central  line,  intersecting  the  line  of  the 
orbit  in  two  points ;  one  being  denoted  by  B,  and  signifying 
September,  or  the  autumnal  equinox ;  and  the  other  denoted  by 
D,  signifying  the  vernal  equinox  ;  which  said  line  so  intersecting 
the  orbit,  may  be  called  the  equinoctial  line. 

The  points  of  intersection  of  the  orbit  by  the  central  line,  may 
be  called  the  zenith  points,  and  the  others  the  equinoctial  points. 

Conceive  tangent  lines  to  the  orbit  to  be  drawn  from  the  equi- 
noctial points,  extending  (parallel  to  each  other)  to  the  fixed  star 
to  be  observed ;  for  it  must  be  understood,  that  notwithstanding 
the  great  diameter  of  the  earth's  orbit,  it  will  not  form  the  base  to 
a  sensible  triangle,  the  vertex  of  which  is  in  a  fixed  star ;  and 
hence  those  tangent  lines  will  be  parallel  to  each  other  to  an  ex- 
actness beyond  detection.  That  tangent  proceeding  from  B, 
being  called  the  B  tangent,  and  that  proceeding  from  C,  the  C 
tangent.  Then  a  body  revolving  in  the  orbit  would,  to  an  ob- 
server at  such  fixed  star,  appear  to  pass  backward  and  forward 
upon  the  equinoctial  line. 

Now  conceive  the  eye  of  the  observer  revolving  in  the  orbit  in 
the  direction  D  A  B  C,  and  to  have  arrived  at  D,  or  the  vernal 
equinox,  when  the  star  will  be  stationary,  and  20"  further  south 
than  it  will  be  when  the  eye  arrives  at  A,  and  40"  further  south 
than  when  the  eye  arrives  at  B.  And,  according  to  Dr.  Brad- 
ley's  theory,  the  reason  why  the  star  appears  stationary  at  D  or 
at  B,  is  because  the  eye  is  then  passing  in  the  line  of  the  eye 
and  object,  or  star.  Hence  at  D  the  star  must  appear  20"  south 
of  the  tangent  line  at  D,  or  20"  out  of  its  true  place  for  days, 
and  at  the  time  and  place  when  the  eye  is  passing  in  a  line 
towards  the  star,  and  not  transverse  to  the  line  of  the  eye  and 
object.  When  the  eye  arrives  at  A  its  motion  is  directly  transverse 
perpendicular  to  the  line  of  the  eye  and  star ;  and  the  star  then 
appearing  20"  further  north  than  when  the  eye  was  at  D,  and 
20"  further  south  than  it  will  appear  when  the  eye  arrives  at  B ; 
and  hence  a  mean  between  the  two  extremes,  it  will  then  at  A 
appear  directly  in  the  zenith,  or  zenith  line ;  and  this  at  the  time 
35 


274  ON    THE    THEORY    OF    THE 

when  its  apparent  motion  northward  is  most  rapid ;  and  when, 
according  to  Dr.  Bradley's  theory,  the  aberration  of  the  star 
should  be  greatest. 

When  the  eye  arrives  at  B  it  views  the  star  20"  to  the  north 
of  the  tangent  line  drawn  from  B  ;  and  there  the  star  again  be- 
comes stationary  for  days,  because  the  motion  of  the  eye  is  in 
the  line  of  the  eye  and  star,  or  object  viewed.  As  the  eye  pro- 
ceeds from  B  to  C,  the  star  returns  southward ;  and  when  the 
eye  arrives  at  C,  the  star  will  again  appear  in  its  true  place,  in 
the  zenith  line,  although  its  apparent  motion  will  then  be  most 
rapid  towards  the  south,  and  the  motion  of  the  eye  will  then  be 
transverse  to  the  line  of  the  eye  and  star ;  and  the  motion  of  the 
star  will  continue  southward  till  the  eye  again  arrives  at  D, 
when  it  will  again  become  stationary  at  20"  out  of  its  true  place  ; 
when  it  will  commence  going  north  again. 

Now  it  would  seem  as  though  no  further  explanation  need 
be  made  to  convince  the  most  casual  observer  of  the  entire  ab- 
surdity of  Dr.  Bradley's  theory.  For  I  will  ask  whether  any  one 
can  reconcile  the  Doctor's  theory  with  the  fact,  that  when  the 
motion  of  the  eye  transverse  to  the  line  of  the  eye  and  star  is 
greatest,  the  star  should  be  in  its  true  place ;  and  when  the  requi- 
site motion  of  the  eye  for  producing  an  aberralion  is  nothing, 
because  it  is  in  a  line  of  the  eye  and  object,  the  aberration  of  the 
star  should  not  only  be  the  greatest,  but  continue  so  for  days, 
namely,  while  the  star  should  appear  stationary  ? 

But  again,  —  the  amount  of  aberration  at  D  or  at  B  is  only 
8  I -4th  minutes  in  time.  Hencevif  the  eye  were  to  remain  still, 
or  were  to  be  passing  in  the  line  of  the  eye  and  star  for  that 
length  of  time,  the  star  must  necessarily  come  to  its  true  place, 
even  upon  Dr.  B.'s  theory.  But  at  D  or  B  the  eye  has  no  requisite 
motion  for  producing  any  aberration  of  the  star  for  days,  and 
yet  the  star  stands  out  south  atD  and  north  at  B,  for  all  this  time. 

But  when  the  eye  comes  to  A  or  to  C,  the  star  appears  in  its 
true  place,  being  a  mean  between  its  north  and  south  declination, 
notwithstanding  the  motion  of  the  eye  is  then  transverse  to  the 
line  of  the  eye  and  star ;  and  as  the  motion  of  the  star  is  then  the 
greatest,  then  according  to  the  Doctor's  theory,  that  if  light  be 
progressive,  and  this  be  the  cause  of  the  apparent  motion  of  the 
star,  its  greatest  aberration  should  be  when  the  eye  is  at  A  or  C ; 
but  it  is  directly  the  reverse,  and  goes  as  far  as  a  theory  can  to 
prove  that  light  is  not  progressive  in  time. 

This  would  seem  to  indicate  that  were  the  motion  of  the  eye 
to  cease  in  any  part  of  the  orbit,  the  star  would  continue  to  re- 
main at  the  place  where  it  appeared  when  the  motion  of  the  eye 
ceased  ;  and  so  doubtless  would  be  the  fact ;  for  it  is  too  obvious 
that  the  aberration  of  light,  upon  Dr.  Bradley's  theory,  is  not  the 


PROGRESSIVE    MOTION    OF    LIGHT.  275 

cause  of  the  phenomena,  which  doubtless  depends  on  a  much 
more  natural,  simple  and  beneficial  cause,  which  I  may  hereaf- 
ter venture  to  suggest. 

Perhaps  the  only  rational  way  of  accounting  for  the  error 
which  those  have  fallen  into  who  have  undertaken  to  ex- 
plain the  Doctor's  theory,  is,  by  supposing  that  they  erred  as  to 
what  places  or  points  of  the  orbit  the  star  would  be  seen  in  its 
true  place ;  and  we  must  conclude  that  they  made  the  true  place 
where  the  star  would  appear  when  stationary;  for  three  days 
were  allowed  for  it  to  come  to  its  true  place,  when  at  most  but 
a  few  minutes  would  be  required  ;  not  considering  that  at  those 
points  where  the  star  appears  stationary,  lines  drawn  to  the  two 
apparent  places  of  the  star  from  those  points  would  form  an 
angle  of  40" ;  and  that  at  A  and  C  the  star  from  both  points 
would  appear  in  the  same  point  in  the  heavens,  and  that,  too,  in 
the  mean  and  the  true  place. 

But  it  will  be  perceived  that  the  star  appears  in  its  true  place 
only  when  the  motion  of  the  eye  is  in  a  direct  transverse  direction 
to  the  line  of  the  eye  and  star ;  and  that  in  no  other  parts  of  the 
orbit  will  the  eye  perceive  the  star  in  its  true  place,  going  upon 
the  true  theory,  whatever  that  may  be. 

And  hence  I  think  it  clear  that  if  the  phenomena  observed  by 
Dr.  Bradley  in  respect  to  the  apparent  motion  of  the  fixed  stars, 
prove  anything,  they  prove  directly  the  opposite  of  what  has  gen- 
erally been  supposed;  and  I  think  it  conclusively  shown  that 
the  motion  of  the  eye,  whether  it  be  much  or  little,  has  nothing 
to  do  in  accounting  for  the  phenomena  observed  by  Dr.  Bradley; 
and  if  any  have  supposed  that  the  star  appeared  in  its  true  place 
at  B  and  at  D,  let  them  attempt  to  extricate  themselves  from 
such  a  dilemma. 

A  nutation  of  the  poles  of  the  earth  is  sometimes  defined  to 
be  "  a  kind  of  tremulous  motion  of  the  axis  of  the  earth,  by 
which  in  its  annual  revolution,  it  is  twice  inclined  to  the  ecliptic, 
and  as  often  returns  to  its  former  position  "  —  and  so  far  as  the 
actual  poles  of  the  earth  partake  of  a  circular  motion,  or  revolu- 
tion about  the  mean  poles  of  the  earth,  such  revolution  is  found 
to  be  in  the  order  of  the  signs. 

Dr.  Bradley  found  in  his  critical  observations  that  the  stars 
which  he  observed,  apparently  had  their  most  rapid  motion  near 
the  winter  and  summer  solstices,  and  their  least  apparent  motion 
near  the  vernal  and  autumnal  equinoxes ;  and  that  a  star  sit- 
uated near  the  solstitial  colures,  would  have  a  greater  apparent 
motion  either  north  or  south,  than  those  situated  near  the  equi- 
noctial colures.  The  Dr.  also  found,  that  it  was  not  a  general 
law  of  the  apparent  motion  of  the  stars  that  they  would  be 


276  PROGRESSIVE    MOTION   OP    LIGHT. 

farthest  north  or  south  when  the  sun  was  about  the  equinoxes ; 
but  that  such  only  held  good  in  respect  to  those  stars  that  were 
near  the  solstitial  colure  ;  nevertheless,  any  star  would  become 
apparently  stationary  (being  farthest  either  north  or  south)  when 
it  passed  over  his  zenith  about  six  o'clock  in  the  morning,  or 
six  o'clock  in  the  evening ;  being  farthest  south  when  it  passed 
at  six  o'clock  in  the  morning ;  and  farthest  north  when  it 
passed  at  six  o'clock  in  the  evening ;  the  apparent  motion  of 
any  star  tending  the  same  way  when  it  passed  his  instrument 
about  the  same  hour  of  the  day  or  night,  with  the  exception  of 
some  small  variation. 

Now  notwithstanding  all  this  evidence,  the  Dr.  says  that  on 
examining  and  comparing  his  evidence,  and  having  pretty  well 
satisfied  himself  as  to  the  general  laws  of  the  phenomena,  he 
then  endeavored  to  find  out  the  cause  of  them ;  that  he  was 
already  convinced  that  the  apparent  motion  of  the  stars  was  not 
owing  to  a  nutation  of  the  earth's  axis.  But  I  am  certainly  at 
a  loss  to  know  from  what  cause  his  conviction  arose  —  inasmuch 
as  all  the  phenomena  detailed  or  set  forth  would  seem  to  refer 
the  nutation  of  the  earth's  axis  and  the  apparent  motion  of  the 
stars  to  the  same  cause. 

But  the  Dr.  says,  at  last  he  conjectured  the  cause  &c.,  which 
conjecture  has  ever  since  been  received  and  acted  upon  as  the 
true  hypothesis.  But  it  would  seem  as  though  most  philos- 
ophers with  all  this  evidence  before  them,  would  have  con- 
jectured that  one  and  the  same  cause  produced  the  nutation  of 
the  earth's  axis,  the  precession  of  the  equinoxes,  and  the  appa- 
rent motion  of  the  fixed  stars ;  not  however,  unless  they  con- 
ceived the  solar  system  to  be  governed  by  some  immutable  law 
of  order,  in  lieu  of  being  governed  by  fortuity,  or  the  Newtonian 
theory  of  universal  gravity,  which  would  indeed  very  illy  ac- 
count for  those  phenomena. 

But  if  we  conceive  the  solar  system  to  be  governed  by  laws 
of  order,  those  phenomena,  viz.  the  nutation  of  the  poles,  the 
precession  of  the  equinoxes,  and  the  apparent  motion  of  the 
stars  observed  by  Dr.  Bradley,  would  seem  to  be  the  direct  and 
natural  results  of  such  law,  and  some  of  the  most  happy  eluci- 
dations of  the  law.  In  such  case  there  would  be  no  lunar  nu- 
tation, but  the  whole  would  be  solar  nutation  ;  nor  would  it  be 
said  as  is  now  said,  that  physical  astronomy  has  made  known  a 
solar  nutation,  but  that  it  is  not  sufficient  to  be  detected  by  ob- 
servation ;  but  if  a  law  of  order  for  the  solar  system  were 
known  and  acknowledged,  it  would,  dictate  to  us  that  such  phe- 
nomena must  necessarily  occur ;  hence  on  observing  their  occur- 
rence there  would  be  no  difficulty  in  assigning  the  true  cause. 


APPENDIX. 


I  have  thought  proper,  in  a  short  appendix,  to  add  something 
further  in  reference  to  the  quadrature  and  its  general  application  ; 
as  also  some  few  additional  remarks  in  reference  to  the  law  of 
gravity. 

As  heretofore,  the  p  whose  M  a  and  C  are  conceived  to  be 
adverse,  is  denoted  by  figure  1  at  the  right  of  its  respective  deno- 
tations ;  and  the  p  whose  B  and  C  are  conceived  to  be  adverse, 
and  whose  M  a  and  D  coincide  in  %,  is  denoted  by  figure  2  at 
the  right  of  its  respective  denotations. 

M  a  and  B  of  the  p  of  twice  the  sides  of  the  p  2  occupy  the 
identical  places  that  C  and  D  of  the  p  do  whose  A  and  B  coin- 
cide in  £,. 

C  and  B  of  the  p  2  occupy  the  identical  places  that  B  and  D 
do  of  the  p  whose  C  is  in  ^. 

M  a  and  C  of  the  p  1  occupy  the  identical  places  that  A  and 
M  a  do  of  the  p  whose  C  is  in  J^. 

The  adverse  of  A  of  any  given  p  is  a  mean  between  2,  and  E 
of  a  like  /?,  and  is  consequently  =  m  2  when  the  area  is  =  E. 

The  adverse  of  M  a  of  any  given  p  is  a  mean  between  ^  an(i 
G  of  a  like  p,  and  is  consequently  =  m  2  when  the  area  is  =  G. 

The  adverse  of  a  mean  between  A  and  M  o,  namely,  the  ad- 
verse of  C,  is  a  mean  between  ^  and  E  of  twice  the  sides,  and 
is  consequently  =  m  2  when  the  area  is  =  E  of  twice  the  sides. 

The  mean  place  of  M  a  is  a  mean  between  M  a  and  C  of  a  p 
of  half  the  sides. 

The  mean  place  of  D  is  a  mean  between  B  of  the  same  p,  and 
D  of  a  p  of  half  the  sides. 

The  square  of  the  mean  place  of  M  a  is  a  mean  between  the 
square  of  the  true  place,  and  the  square  of  E  of  a  p  of  twice  the 
sides. 


278  APPENDIX. 

The  square  of  the  mean  place  of  D  or  of  C,  is  a  mean  be- 
tween the  square  of  the  true  place  of  D,  and  the  square  of  the 
true  place  of  C ;  and  the  square  of  the  mean  place  of  A  is  a  mean 
between  the  square  of  the  true  place  of  D  of  half  the  sides,  and 
the  square  of  the  true  place  of  C  of  half  the  sides. 

The  product  of  Ma  4  by  C,  is  jl  above  the  square  of  "Ma  8, — 
the  product  of  M  a  8  by  C  8,  is  L  above  the  square  of  M  a  16, — 
the  product  of  M  a  16  by  C  16,  is  ^  above  the  square  of  M  a  32, — 
and  so  on,  until  the  product  of  Ma  1  by  C  1,  is  the  square  of 
M  a  2.  And  if  we  attempt  to  continue  a  like  progression,  we 
shall  find  a  uniformity  which  has  not  previously  existed;  for 
the  product  of  Ma  2  by  C  2,  is  =  the  square  of  M  a  of  twice 
the  sides,  and  such  will  continue  to  be  the  case,  however  far  we 
attempt  to  continue  the  progression.  Hence,  the  product  of  M  a 
by  C,  can  never  be  below  the  square  of  Ma  of  a p  of  twice 
the  sides ;  and  hence,  from  the  p  1  onward,  our  progression  is 
futile  and  void. 

The  product  of  M  a  by  D,  is  always  =  the  product  of  B  by 
C ,  or  of  B  of  the  given  p  by  A  of  &p  of  twice  the  sides ;  and  as  in 
the  circle,  M  a  actually  coincides  with  C,  hence,  our  conception 
necessarily  is,  that  in  the  circle,  M  a  coincides  with  A  of  twice 
the  sides. 

When  D  is  conceived  to  be  JL  below  Ma,  such  D  is  the 
adverse  of  the  popular  D,  or  B  of  the  circle ;  and  such  M  a  is 
the  adverse  of  the  popular  Ma  or  C  of  the  circle;  and  such  Ma 
is  then  a  mean  between  D  and  B  of  the  same  J9,  and  is  conse- 
quently =  B  of  a  p  of  twice  the  sides. 

B  2  is  conceived  to  be  the  farther  of  two  means  from  B  1,  to 
^  —  and  the  popular  B  of  the  circle,  is  the  farther  of  two  means 
from  B  2  to  ^. 

„£,  is  the  farther  of  two  means  from  A  to  B,  hence,  when  A 
and  B  coincide  in  ^,  such  point  is  as  well  the  farther  of  two  means 
from  A  to  B  of  twice  the  sides,  as  from  A  to  B  of  half  the  sides. 

If  we  conceive  a  p  of  twice  the  sides  of  that  whose  A  and  B 
coincide  in  ^  ,  we  necessarily  conceive  M  a  of  such  p  to  be  2_ 
below^,  or  in  the  place  of  C  1,  or  of  A  2 ;  and  also,  that  A  of 
such  p  is  j?.  above  ^,  or  in  the  place  of  M  a  1 ;  and  that  C  of 
such  p,  is  in  ^,  and  B  of  such  p,  will  be  conceived  to  be  J_ 
below  ^  and  D  JL  above  ^.  If  in  the  p  1  we  conceive  1  to 
po^s  ss  the  same  value  as  in  the  p  8  and  still  conceive  B  to  con- 
tinue _L  above  M  a  and  D  to  continue  to  be  J_  above  C,  and  attempt 
to  continue  the  progression  beyond  the  p  1  and  until  B  and  D  co- 
incide J_  above  Ma  and  C,  we  necessarily  place  B  and  D  final 
|  above  ^,  and  M  a  and  C  final  f  below  ^,.  But,  as  before 
remarked,  from  the  p  1  onward,  we  do  not  displace  Ma  from  its 
constant  mean  position  between  M  a  and  C  of  p  of  half  the  sides. 


.        APPENDIX.  279 

So  also,  if  we  conceive  the  progression  to  be  continued  after 
A  and  B  coincide  in  ^,  we  shall  arrive  at  the  same  conceived 
result,  namely,  shall  place  B  and  D  final  |  above  J^,  and  M  a 
and  C,  final  f  below  J^ ;  but  it  is  manifest  that  such  continued 
progression  will  be  by  an  apparent  retrograde  movement  of  the 
denotations  from  that  p  which  is  conceived  to  have  twice  the 
sides  of  that  whose  A  and  B  are  in  J^,.  That  is,  when  C  is  in 
^,  we  conceive  M  a  to  be  1.  below  M  a  when  A  is  in  £, ;  and 
if  B  continues  to  be  _L  above  M  a,  and  if  B  of  the  circle  is  1_ 
above  M  a  of  the  circle  (if  such  _L  then  has  any  value,  whatever 
value  it  may  then  have,)  the  final  B  and  D  must  be  ^  (of  such 
value)  above  J^,  and  M  a  and  C  final  must  be  f  (of  such  value) 
below  J^ ;  but  if  in  the  circle  there  is  no  value  to  _L,  then  M  a 
and  D  of  the  circle  are  in  J^. 

In  such  case,  A  in  J^,  must  be  conceived  to  be  A  of  twice  the 
sides  of  Ma  of  the  circle;  and  consequently,  we  must  conceive 
A  of  half  the  sides  of  A  in  J^,  to  be  in  reality  the  utmost  A, 
and  it  is  manifest  that  we  can  only  conceive  such  A  of  half  the 
sides  to  be  =  Ma  1.  And  the  product  of  Ma  1  by  C  1  is  the 
square  of  the  true  place  of  Ma  2,  and  of  the  mean  place  of.  Ma 
of  four  times  the  sides  of  the  p  1,  and  the  like  may  be  conceived 
in  respect  to  the  circle.  Hence,  in  ^,  the  true  and  mean  places 
of  M  a  coincide,  as  they  do  in  the  p  4. 

Hence  M  a  in  J^,,  nas  risen  to  a  mean  between  M  a  and  C  of  a  p 
of  half  the  sides ;  and  the  popular  methods  do  not  attempt  anything 
farther.  M  a  S  is  1_  below  a  mean  between  M  a  and  C  of  a  /;  of 
half  the  sides;  hence,  _L,  which  in  the  p  8  denotes  a  mean  be- 
tween Ma,  and  a  mean  between  Ma  and  C  of  a  p  of  half  the 
sides,  is  equated  to  JL  in  the  p  2  ;  and  no  one  expects  that  Ma 
will  either  rise  above  a  mean  between  M  a  and  C  of  a  p  of  half 
the  sides,  or  that  the  law  of  progression  will  so' far  change,  that 
Ma  will  continue  (while  the  progression  continues)  to  be  a 
mean  between  M  a  and  C  of  a  p  of  half  the  sides. 

M  a  then  coincides  with  M  a  and  C  of  half  the  sides,  at  the 
time  and  place  when  it  becomes  a  mean  between  them,  and  then 
it  necessarily  coincides  with  C  of  the  same  jt?,  thus  making  the 
four-fold  chord  of  the  periphery  of  the  circle. 

Thus,  by  the  law  of  polygonal  progression,  M  a  cannot  be 
below  J^  unless  it  be  above  a  mean  between  M  a  and  C  of  half 
the  sides ;  and  by  any  method,  it  is  supposed  that  when  M  a 
becomes  a  mean  between  M  a  and  C  of  half  the  sides,  it  coin- 
cides with  them. 

Suppose  then,  that  the  difference  between  Mai  and  B  1,  or 
between  M  a  2  and  B  2,  to  have  even  been  increased  in  value 
from  the  p  S,  according  to  the  popular  increase,  and  then  attempt 
to  obtain  M  a  of  twice  the  sides  of  the  p  1  in  the  popular  man- 


280  APPENDIX. 

ner, — that  is,  by  dividing  the  product  of  Ma  1  by  C  1,  by  half 
the  sum  of  M  a  1  and  C  1. 

In  such  case,  M  a  of  a  p  of  twice  the  sides  of  the  p  1  will  be 
proportionally  as  far  below  ^  as  an  equal  mean  between  Mai 
and  C  1  is  above  J^ ;  that  is,  if  M  a  1  and  C  1  are  not  in  ^,  by 
such  process,  M  a  of  twice  the  sides,  will  be  below  J^,  and  con- 
sequently, must  be  above  a  mean  between  Mai  and  C  1  — 
which  cannot  be  the  case  —  neither  can  such  Ma  be  below  ^ 
for  when  M  a  and  C  are  adverse,  M  a  of  twice  the  sides  is  neces- 
sarily in  J^.  Thus,  showing  direct  error,  in  the  popular  mode  of 
conducting  the  progression. 

As  C  is  a  mean  between  M  a  of  the  same  p  and  C  of  half 
the  sides,  if  after  obtaining  the  popular  M  a  2,  we  then  attempt 
to  obtain  C  2,  we  shall  obtain  it  proportionally,  half  as  far  too 
low  as  the  popular  M.  a  2  will  be ;  and  it  will  be  manifest  to 
any  one,  that  the  error  will  continue  to  increase  during  the  pro- 
gression. But  as  M  a  cannot  become  a  mean  between  M  a  and 
C  of  a  p  of  half  the  sides  till  it  coincides  with  them,  and  as 
M  a  in  ^  is  a  mean  between  M  a  and  C  of  a  p  of  half  the  sides, 
hence,  Ma  in  ^  coincides  with  Ma  and  C  of  half  the  sides. 

And  as  in  the  case  of  M  a  in  J^,,  in  which  M  a  has  then  risen 
to  a  mean  between  M  a  and  C  of  a  p  of  half  the  sides,  so  D  in 
£,  has  fallen  to  a  mean  between  B  of  the  same  JP,  and  D  of  a  p 
of  half  the  sides.  Hence,  in  J^,  the  relative  situation  of  D  will 
have  altered  in  respect  to  a  mean  between  B  of  the  same  />,  and 
D  of  a  p  of  half  the  sides,  from  what  it  is  in  the  p  8,  proportion- 
ally half  as  much  as  the  relative  situation  of  Ma  in  J^,  in 
respect  to  a  mean  between  M  a  and  C  of  half  the  sides  will 
have  shifted  from  what  it  is  in  the  p  8. 

Hence,  in  ^  the  mean  places  of  both  M  a  and  D  will  be  their 
true  places  ;  nor  can  D  by  rising  above  J^,  fall  below  a  mean 
between  B  of  the  same  jo,  and  D  of  a  p  of  half  the  sides,  any 
more  than  M  a  by  falling  below  J^,  can  rise  above  a  mean  be- 
tween M  a  and  C  of  a  p  of  half  the  sides.  Consequently,  in  J^ 
the  progression  is  ended  in  respect  to  M  a  and  D,  and  hence,  inj^ 
M  a  and  D  become  M  a  and  I)  of  the  circle,  which  are  the  final 
quantities  sought.  And  as  the  mean  place  of  D  is  a  mean 
between  D  and  C,  hence,  when  the  true  and  mean  place  of  D 
coincide,  D  is  then  =  C,  hence,  when  D  is  in  J^,  C  is  in  ^,  as 
also  Ma. 

Such  then,  gives  the  true  or  actual  ratio  between  the  diameter 
and  circumference  of  the  circle,  the  proportions  of  which  may  be 
expressed  in  the  powers  and  roots  of  definite  numerical  quanti- 
ties ;  making  the  circle  in  its  infinite  use  and  application  a 
pleasant,  rather  than  irksome  task. 

I  will  here  notice  a  few  additional  facts,  by  way  of  application. 


APPENDIX.  281 

The  fifth  power  of  M  a  of  the  circle,  is  d  of  the  circle,  whose 
circumference  is  1,  and  if  such  be  an  orbit,  the  period  of  the 
planet  will  be  =  one  sixteenth  of  the  circumference. 

If  the  circumference  of  an  orbit  be  4,  the  period  is  one  eighth 
of  the  circumference ;  thus,  if  the  circumference  be  quadrupled, 
the  ratio  of  the  period  to  the  circumference  is  reduced  one  half. 
So  when  the  circumference  is  16,  the  square  of  the  period  is  = 
to  the  circumference ;  and  this  occurs  when  the  diameter  of  the 
orbit  is  half  the  square  of  the  circumference  of  the  prime  circle. 

If  the  diameter  of  the  inscribed  circle  of  a  given  /?,  be  =  to 
the  square  of  the  circumference  of  a  like  major/?,  the  circumfer- 
ence of  such  polygon,  will  be  the  third  power  of  the  circumfer- 
ence of  such  major  p ;  hence,  if  the  diameter  of  the  circle  be 
10.07936  (square  of  the  circumference  of  the  prime  circle)  the 
circumference  of  such  circle  will  be  32. 

The  area  of  a  circle  is  always  twice  the  square  of  IT  below  m  ; 
so  if  m  of  the  circle  be  IT  above  two  thirds  of  1,  (namely,  when 
m  of  the  circle,  is  .839946)  the  area  is  .888,  ad  infinitum ;  which 
area  is  =  the  area  of  the  p  6,  when  m  of  such  p  6  is  =  B  6  ;  as 
.888  is  the  area  of  the  circle  whose  m  is  two  thirds  of  b  a  of  the 
circle,  so  also,  it  is  half  the  area  of  the  p  4  whose  m  is  two  thirds 
of  b  a  of  the  p  4,  and  is  twice  the  area  of  the  p  4  whose  m  is 
two  thirds  of  da  of  the  p  4. 

When  the  area  of  the  circle  is  .888,  ad  infinitum,  m  of  such 
circle  is  the  cube  root  of  two  thirds  of  the  area,  and  is  the  recip- 
rocal of  the  cube  root  of  the  square  of  M  a  3.  So  four  thirds  of 
M  a  of  the  circle  is  b  of  the  circle  whose  area  is  .888,  ad  infinitum, 
and  four  thirds  of  M  a  of  the  p  4  is  b  of  the  square  whose  area 
is  .888,  ad  infinitum. 

Three  times  the  square  of  M  a  4  is  the  reciprocal  of  one-third 
of  M  a  4,  and  three  times  the  square  of  M  a  of  the  circle  is  the 
reciprocal  of  two-thirds  of  M  a  of  the  circle,  or  of  the  bulk  of  the 
prime  sphere.  So  the  square  of  M  a  4  is  half  of  b  a  4,  and  the 
square  of  M  a  of  the  circle  is  half  of  b  a  of  the  circle  ;  but  such 
is  not  the  case  with  any  p  between  the  p  4  and  circle. 

The  diameter  of  any  regular  solid  is  the  diameter  of  its  in- 
scribed sphere ;  and  four  times  the  square  of  M  a  3  is  the  third 
power  of  the  reciprocal  of  the  bulk  of  the  sphere  whose  diameter 
is  1,  namely,  6.75  is  the  third  power  of  1.889881 ;  and  when  the 
diameter  of  the  sphere  is  1.889881  its  surface  will  be  12. 

The  third  power  of  the  entire  surface  of  any  prime  prism  -=-  by 
the  third  power  of  the  lateral  surface  of  such  prism  gives 
3.375,  or  twice  the  square  of  M  a  3.  And  of  course,  the  third  pow- 
er of  the  surface  of  the  prime  cube  -r-  by  the  third  power  of  the  sur- 
face of  the  prime  sphere,  gives  four  times  the  square  of  M  a  3. 
36 


282  APPENDIX. 

When  the  diameter  of  any  given  prism  is  =-  d  a  of  a  like  pj 
the  bulk  of  such  prism  will  be  the  square  of  the  diameter,  for 
the  reason  that  bulk  flows  by  a  triplicate  ratio  to  that  of  diame- 
ter, or  any  other  linear  measure  of  the  given  solid ;  and  in  such 
case,  the  surface  will  be  six  times  the  diameter,  for  the  reason 
that  surface  flows  by  a  duplicate  ratio  to  that  of  diameter. 

We  know  but  little  or  nothing  of  the  mode  of  operation  of  that 
power  or  force  (better  expressed  by  the  word  attraction  than  by 
any  other)  which,  like  a  messenger  of  Omnipresence,  operates 
throughout  the  solar  system,  (and  by  analogy,  throughout  the 
universe,)  in  causing  the  gravity  or  ponderosity  of  the  bodies  of 
aggregate  matter  which  compose  the  solar  system,  and  also  in 
controlling  their  motions  during  their  eternal  rounds. 

The  phenomena,  however,  caused  by  the  force  of  attraction, 
enable  us  to  contemplate  many  of  its  physical  effects,  and  to  in- 
vestigate and  determine  their  ratios  and  proportions  by  means  of 
the  powers  and  roots  of  numbers,  whatever  the  number  of  ele- 
ments may  be  which  enter  into  the  calculation,  as  that  of  time, 
space,  distance,  force,  convergency,  motion,  &c. 

From  the  phenomena  observed  in  the  solar  system,  which  are 
manifestly  caused  by  the  power  or  force  of  attraction,  we  proper- 
ly conceive  the  sun  to  be  situated  in,  or  to  be  the  centre  of  at- 
traction for  the  solar  system  ;  and  hence,  that  all  bodies  of  matter 
in  the  solar  system,  which  revolve  around  the  sun,  are  attracted 
and  retained  in  their  orbits  by  the  sun.  Nevertheless,  those 
bodies  which  revolve  around  the  sun  are  observed  to  possess  the 
power  of  attraction  ;  hence  each  planet  having  satellites,  attracts 
and  retains  them  in  their  orbits. 

But  a  question  which  may  yet  require  a  further  and  a  better 
consideration  of  philosophers  and  astronomers,  is,  whether  a  like 
quantity  of  matter,  wherever  situated  in  the  solar  system,  possess- 
es a  like  power  of  attraction,  or  a  like  power  of  attraction  for  an- 
other body  of  matter  ;  or  whether  the  power  of  attraction  inci- 
dent to  a  body  of  matter  situated  in  the  solar  system,  may  not 
depend  upon  its  distance  from  the  sun,  or  centre  of  attraction  of 
the  system  ?  If  the  power  of  attraction  of  a  given  quantity  of 
matter  be  the  same,  wherever  situated,  then  Jupiter  must  be  far 
less  dense  than  the  earth ;  but  if  the  power  of  attraction  of  a 
given  quantity  of  matter  in  the  solar  system  be  inversely  as  the 
distance  of  such  matter  from  the  sun,  or  centre  of  attraction  of 
the  system,  then  the  density  of  the  matter  of  which  Jupiter  and 
the  earth  are  composed  may  be  considered  as  being  equal,  or 
substantially  so.  And  in  such  case  we  should  say  that  the  pow- 
er of  attraction  of  a  given  quantity  of  matter  in  the  solar  system 
is  inversely  as  its  distance  from  the  sun,  or  centre  of  attraction  of 


APPENDIX.  283 

the  solar  system.  And  that  this  hypothesis  is  the  true,  would 
seem  to  be  well  established  from  the  observations  and  calcula- 
tions of  some  of  the  most  astute  mathematicians  and  astrono- 
mers, who  have  found  the  law  to  be  so  general,  uniform  and 
exact,  as  to  construct  nice  astronomical  tables  upon  its  cor- 
rectness. 

But  it  must  be  manifest  to  any  person,  that  the  theory  of  the 
different  densities  of  the  planets  can  have  no  other  support  in 
favor  of  it,  than  that  of  the  Newtonian  hypothesis,  that  all  matter, 
wherever  situated,  is  equally  endued  with  an  innate  or  inherent 
principle  of  attraction  ;  which  theory,  on  examination,  will  per- 
haps be  found  not  to  be  supported  or  sustained  by  a  single  fact 
or  phenomenon  found  to  exist  in  the  universe.  Nor  is  it  at  all 
certain  that  the  atheistical  tendency  of  such  theory  was  relieved 
by  the  vague  conjecture  that  the  Almighty  originally  pro- 
jected the  heavenly  bodies  in  tangents  to  their  orbits,  and 
thereby  placed  them  under  the  eternal  control  of  this  original, 
innate  power  of  attraction ;  for  such  hypothesis  only  presupposes 
that  attraction  had  previously  aggregated  the  masses  of  matter, 
each  of  which  was  waiting  for  the  impulse  to  be  given.  Never- 
theless, this  naked  conjecture  or  hypothesis,  wholly  unsustained 
or  supported  by  any  physical  law  or  phenomenon  with  which 
we  are  acquainted,  is  too  generally  conceived  to  be  a  full  and 
ample  belief  in,  or  acknowledgement  of  the  wisdom  and  power 
of  Deity.  But  I  must  dissent  from  those  philosophers  who  at- 
tempt to  inculcate  such  notions  by  mingling  them  with  the  do- 
ings of  a  deity  of  their  own  making.  Such  is  certainly  not  an 
exposition  or  explanation  of  natural  philosophy,  but  is  emphati- 
cally what  Newton's  admiring  biographer  alleges,  —  a  creation 
of  natural  philosophy. 

It  is  not,  perhaps,  the  better  way,  in  the  development  of  nat- 
ural philosophy,  to  reject  everything  as  being  vulgar  which  falls 
under  common  observation,  and  strike  at  some  bold  and  original 
hypothesis,  as  the  foundation  to  which  everything  must  be 
squared,  (or  rather  equated,)  in  order  that  one  may  have  the  name 
of  creating  an  entire  system  ;  nor  to  cast  aside  as  useless  all  the 
actual  evidences  we  possess,  which  can  aid  us  in  our  investiga- 
tions, even  though  such  evidence  may  (by  certain  philosophers) 
have  been  used  in  a  whimsical  or  futile  manner,  by  attempting 
to  prove  more  by  it  in  respect  to  the  phenomena  of  nature  than 
is  warranted  by  true  philosophy.  Nevertheless,  the  most  com- 
mon observer  must  perceive  some  analogy  between  the  law  of 
attraction,  as  operating  in  the  solar  system,  and  the  operation  of 
the  laws  of  magnetism,  electricity,  galvanism  and  electro-mag- 
netism, which  we  are  able  to  elicit  for  our  use  and  edification 


284  APPENDIX. 

from  the  great  store-house  or  laboratory  of  nature ;  which  are 
perhaps  but  modifications  of  one  great  physical  law  of  the  uni- 
verse ;  all  tending,  however,  to  induce  the  belief  that  it  is  an 
omnipresent  law,  and  is  not  under  the  necessity  either  to  wait 
for  its  prey,  or  to  reach  for  it. 

The  equal  innate  gravity  of  matter,  wherever  situated,  necessa- 
rily (if  adopted  as  an  axiom)  draws  after  it  the  concomitant  the- 
ory of  the  different  densities  of  the  planets ;  and  if  either  hypoth- 
esis is  true,  they  both  doubtless  are.  So  also  if  either  of  them  be 
untrue,  they  doubtless  both  are.  But  to  me  both  hypotheses 
appear  to  be  wholly  uncalled  for,  and  unsupported  by  any  legiti- 
mate evidence  of  which  we  can  avail  ourselves ;  and  that  in  lieu 
of  the  round-about  process  in  attempting  to  ascertain  the  differ- 
ent densities  of  the  planets  having  satellites,  supposed  to  exist 
in  consequence  of  the  different  attractive  powers  which  they  ex- 
ert upon  their  respective  satellites,  in  proportion  to  the  bulks  or 
magnitudes  of  the  planets,  we  have  only  to  consider  the  quantity 
of  matter  in  the  respective  planets  to  be  as  their  bulks  or  magni- 
tudes, and  that  their  powers  of  attraction,  in  proportion  to  their 
respective  magnitudes,  are  inversely  as  their  respective  distances 
from  the  sun  or  centre  of  gravity  of  the  system  ;  and  especially 
when  we  are  sustained  by  the  evidence,  the  fact,  in  accordance 
with  the  general  and  well  known  law,  that  their  respective  rela- 
tive powers  of  attraction  are  given  by  multiplying  their  respec- 
tive bulks  or  magnitudes  by  the  inverse  of  their  respective  dis- 
tances from  the  centre  of  gravity  of  the  solar  system ;  for  here  is 
a  fact  found,  according  with  a  law  found  to  be  uniform  and 
universal,  that  the  squares  of  the  periods  of  the  planets  are  as 
the  cubes  of  their  mean  distances  from  the  sun. 

I  am  aware  that  the  adoption  of  my  premises  would  abstract 
from  mankind  a  vast  amount  of  admiration  ;  for  "  still  they  gazed, 
and  still  the  wonder  grew,"  that  human  intellect  could  become 
so  sublimed  as  to  be  able  to  weigh  off  the  planets,  and  determine 
their  different  densities,  when  in  reality  the  exploit  supposed  to 
have  been  accomplished,  was  but  a  corollary  flowing  from  one  of 
the  greatest  absurdities  that  was  ever  interwoven  into  a  code  of 
philosophy. 

Let  us  now  examine  a  little  into  the  attractive  powers  of  Ju- 
piter and  the  earth.  The  time  of  the  period  of  the  earth  to  that 
of  Jupiter  is  as  1  to  11.8568 ;  and  the  mean  motion  of  Jupiter  to 
that  of  the  earth  is  as  1  to  2.2802.  The  mean  distance  of  the 
earth  from  the  sun  to  that  of  Jupiter  is  as  1  to  5.1993,  and  the 
cube  or  third  power  of  5.1993  is  the  square  of  11.8568 ;  hence 
the  square  of  the  reciprocal  of  the  rate  of  mean  motion  is  the 
mean  distance. 


APPENDIX.  285 

Hence,  the  mean  attractive  power  of  a  planet  applied  to  its  sat- 
ellite to  retain  it  in  its  orbit,  is  the  square  of  the  mean  motion  of 
the  satellite ;  hence  the  whole  amount  of  motion  of  a  satellite 
during  its  entire  period  is  the  square  root  of  the  whole  amount 
of  the  force  of  attraction  expended  upon  the  satellite  during  the 
period.  Thus  the  rate  of  mean  motion,  or  the  square  root  of  the 
rate  of  mean  force,  multiplied  by  the  time  of  the  period,  will 
give  the  time  of  the  period  of  the  satellite  when  operated  upon 
by  a  different  rate  of  force,  but  at  the  same  distance  from  the  at- 
tracting body. 

According  to  La  Grange,  the  power  of  attraction  of  the  earth 
to  that  of  Jupiter,  is  as  1  to  313.0640,  and  the  square  root  of 
313.064  is  17.6936,  which  of  course  would  be  the  rate  of  mean 
motion  of  a  satellite  revolving  around  Jupiter,  if  the  rate  of  mean 
motion  of  a  satellite  revolving  around  the  earth  at  the  same  dis- 
tance were  1.  The  distance  of  Jupiter's  first  satellite  is  found  to 
be  a  little  greater  from  the  centre  of  Jupiter  than  the  moon  is 
from  the  centre  of  the  earth.  The  period  of  Jupiter's  first  satel- 
lite is  one  day  and  .7639  of  a  day,  namely,  1.7639  days. 

If,  then,  the  time  of  the  period  of  Jupiter's  first  satellite  be 
multiplied  by  the  square  root  of  the  attractive  power  of  Jupiter, 
compared  with  that  of  the  earth,  which  is  assumed  at  1,  namely, 
by  17.6936,  it  gives  31.20974  days,  which  would  be  the  time  of 
the  period  if  such  satellite  were  revolving  around  the  earth  at  the 
same  distance  from  the  centre  that  it  revolves  from  the  centre  of 
Jupiter.  And  the  square  of  the  period  of  the  moon  is  to  the 
cube  of  its  mean  distance  from  the  earth  as  the  square  of 
31.2097  is  to  the  cube  of  the  mean  distance  of  Jupiter's  first  sat- 
ellite from  the  centre  of  Jupiter.  Hence  the  rate  of  mean  motion 
of  Jupiter's  first  satellite  is  as  much  greater  than  it  would  be  if 
revolving  about  the  earth  at  the  same  distance,  as  17.6936  is 
greater  than  1  ;  and  this  calculation  will  be  found  to  rest,  or  to 
be  based  upon  the  principle,  that  if  the  bulk  or  magnitude  of  the 
earth  be  1,  and  its  distance  from  the  sun  be  1,  and  its  attractive 
power  be  1,  the  relative  or  proportional  power  of  attraction  of 
Jupiter,  compared  with  the  earth,  will  be  obtained  by  multiply- 
ing the  bulk  or  magnitude  of  Jupiter,  when  compared  with  the 
earth,  by  its  inverse  distance  from  the  sun  when  compared  with 
the  earth. 

If  we  assume  the  mean  distance  of  the  earth  from  the  sun  at 
95,000,000  of  miles,  and  the  distance  of  the  moon  from  the  earth 
at  237,500  miles,  the  relative  distance  of  the  moon  from  the  earth 
to  that  of  the  earth  from  the  sun  will  be  as  1  to  400.  These  dis- 
tances are  however  assumed  for  the  sake  of  round  numbers,  the 
actual  mean  distance  of  the  moon  from  the  earth  being  general- 
ly stated  at  240,000  miles. 


286  APPENDIX. 

The  periodic  time  of  the  moon  around  the  earth  is  to  that  of 
the  earth  round  the  sun  as  1  to  13.3683.  If  we  then  adopt  the 
foregoing  assumption  as  to  distance,  the  mean  motion  of  the 
moon  in  its  revolution  about  the  earth  will  be  to  that  of  the  earth 
about  the  sun  as  1  to  29.9207,  and  in  such  case,  while  the  square 
of  the  mean  motion  of  the  moon  is  1,  the  square  of  the  mean 
motion  of  the  earth  will  be  895.2482  ;  and  in  such  case  the  rela- 
tive power  of  attraction  of  the  earth  upon  the  moon  to  that  of  the 
sun  upon  the  earth  would  be  as  1  to  895.2482.  But  if  we  as- 
sume the  distance  of  the  moon  from  the  earth  at  240,000  miles, 
the  relative  attraction  of  the  earth  upon  the  moon  to  that  of  the 
sun  upon  the  earth  would  be  as  1  to  876.7402.  Hence  the  force 
of  attraction  exerted  by  the  sun  upon  the  earth  would  be  some 
40,000  times  as  great  as  that  exerted  by  the  moon  upon  the 
earth ;  nor  would  this  power  of  attraction  exerted  by  the  sun  upon 
the  earth  have  a  direct  operation  in  any  wise  upon  the  tides. 

But  I  will  close  this  article  by  a  few  doctrines  relative  to  the 
laws  of  force  and  motion,  with  an  additional  reference  to  the  law 
of  falling  bodies. 

Twice  the  motion  of  a  revolving  body  requires  quadruple  the 
force  of  attraction  to  retain  the  planet  in  its  orbit. 

The  period  of  any  planet  is  the  square  root  of  the  cube  of  its 
mean  distance  from  the  centre  of  gravity. 

If  A  denote  a  planet  at  the  distance  1,  and  B  a  planet  at  the 
distance  2,  the  time  in  which  B  will  revolve  over  a  space  equal 
to  the  circumference  of  A's  orbit,  will  be  =  to  the  square  root  of 
B's  distance,  or  of  the  diameter  of  A's  orbit.  Hence  the  time  in 
which  B  will  revolve  over  a  space  =  to  the  circumference  of  A's 
orbit,  will  be  inversely  as  the  rate  of  the  mean  motion  of  B. 

Hence  the  rate  of  mean  motion  of  a  planet  as  proportioned  to 
the  mean  distance,  is  inversely  as  the  square  root  of  the  mean 
distance ;  or  the  square  root  of  the  mean  distance  is  the  inverse 
of  the  rate  of  mean  motion.  Hence  the  mean  distance  is  a  mean 
proportional  between  the  time  of  the  period  and  the  square  root 
of  the  mean  distance. 

If  D  denote  another  planet  whose  mean  distance  is  4,  the  time 
required  for  D  to  pass  over  a  space  equal  to  the  circumference  of 
A's  orbit  will  be  equal  to  the  square  root  of  D's  distance.  Hence 
if  ever  so  many  planets  revolve  round  the  same  central  force, 
and  the  mean  distance  of  either  of  them  be  assumed  at  unity, 
the  time  required  for  any  other  planet  to  revolve  over  a  space 
equal  to  the  circumference  of  the  orbit  whose  distance  is  1,  will 
be  equal  to  the  square  root  of  the  mean  distance  of  the  planet, 
compared  with  that  whose  distance  is  1.  This,  however,  can 
only  be  true  in  respect  to  centric  orbits. 


APPENDIX.  287 

So  when  the  mean  distance  is  1,  the  rate  of  mean  motion  is 
=  to  the  time  of  the  period.  When  the  distance  is  2  the  rate  of 
mean  motion  is  equal  to  one  fourth  of  the  period.  When  the 
distance  is  3  the  rate  of  mean  molion  is  equal  to  one  ninth  of 
the  period  ;  and  so  on.  Hence,  if  the  force  of  attraction  be  in- 
versely as  the  distance,  it  is  the  square  of  the  mean  motion  ; 
but  if  the  force  of  attraction  be  inversely  as  the  square  of  the  dis- 
tance, it  will  be  the  fourth  power  of  the  mean  motion. 

If  we  take  two  planets  of  a  system,  as  B  and  D,  if  the  distance 
of  B  is  the  square  root  of  the  distance  of  D,  the  period  of  B,  the 
rate  of  motion  of  B,  and  the  rate  of  force  of  B,  will  respectively 
be  the  square  root  of  like  elements  of  D. 

If  the  mean  distance  of  A  be  1,  and  that  of  D  be  4,  if  the  force 
of  gravity  varies  inversely  as  the  distance,  D  will  receive  twice 
the  force  in  the  time  of  its  period  that  A  will  in  the  time  of  its 
period.  But  if  the  force  of  gravity  varies  inversely  as  the  square 
of  the  distance,  D  will  receive  but  half  the  force  in  the  time  of  its 
period  that  A  will  in  the  time  of  A's  period. 

The  mean  distance  of  a  planet  multiplied  by  the  square  root 
of  the  mean  distance,  gives  the  period  ;  and  the  reciprocal  of  the 
period  of  a  planet  is  the  product  of  the  rate  of  motion  by  the  rate 
of  force.  Hence  when  the  circumference  of  the  orbit  is  16,  and 
the  time  of  the  period  is  4,  the  radius  of  the  orbit  or  the  distance  is 
then  the  third  root  of  16,  and  the  square  root  of  such  radius  is 
the  third  root  of  four ;  and  the  product  of  such  radius  by  its 
square  root  gives  the  time  of  the  period,  or  one  fourth  of  the  cir- 
cumference of  the  orbit.  But  when  the  period  is  8,  the  product 
of  radius  by  the  square  root  of  radius,  gives  as  well  the  diame- 
ter as  the  period.  Or  the  version  maybe  thus,  —  the  cube  root 
of  the  period  multiplied  by  the  cube  root  of  the  square  of  the 
period  gives  the  period. 

By  the  popular  determination  of  the  quadrature,  when  the 
period  of  the  planet  is  =  one  fourth  of  the  circumference  of  the 
orbit,  such  period  is  2_  below  4 ;  for  when  the  period  is  4  by  the 
popular  mode,  one  fourth  of  the  circumference  of  the  orbit  is  f 
below  4.  Hence  by  any  system  of  the  quadrature,  when  the  peri- 
od of  the  planet  is  =  one  fourth  of  the  circumference  of  the  orbit, 
the  time  of  the  period  will  be  the  third  power  of  twice  M  a  of 
the  circle. 

So  by  any  system,  when  the  period  is  =  one  fourth  of  the  cir- 
cumference of  the  orbit,  the  circumference  of  the  orbit  is  half  of 
the  third  power  of  the  circumference  of  the  circle  whose  diam- 
eter is  1.  . 

Nevertheless,  by  the  popular  determination  of  the  quadrature, 
if  the  period  of  a  planet  be  made  commensurable  with  the  di- 
ameter of  the  orbit  in  the  powers  and  roots  of  definite  quan. 


288  APPENDIX. 

titles,  it  will  not  be  commensurable  with  the  circumference  of 
the  orbit,  or  with  any  part  of  it ;  and  if  the  period  be  made 
commensurable  with  the  circumference  of  the  orbit,  or  with  any 
part  of  it,  (namely,  with  the  path  in  which  the  planet  travels,)  it 
will  not  be  commensurable  with  the  diameter.  But  by  the  true 
determination  of  the  quadrature  of  the  circle,  the  period  of  a 
planet  is  as  well  commensurable  with  the  circumference  of  its 
orbit  as  with  its  diameter. 

I  have  heretofore  alleged,  that  by  the  true  law  of  gravity  or 
attraction,  the  earth  would  exert  quadruple  the  amount  of  the 
force  of  attraction  upon  the  moon,  that  would  be  exerted  by  the 
Newtonian  law ;  and  hence  the  true  law  would  be  just  what 
seemed  to  be  required  by  Clairaut,  Euler  and  others,  for  the 
accounting  for  the  motion  of  the  moon's  apogee,  upon  the  princi- 
ple of  attraction.  So  by  the  true  law  of  gravity,  any  planet  in 
the  solar  system  will  be  attracted  towards  the  sun  by  quadruple 
the  force  that  it  would  be  by  the  Newtonian  law. 

And  lest  any  one  should  not  readily  apprehend  the  truth  of 
the  allegation,  I  will  shortly  explain,  nevertheless,  some  part  of 
the  explanation  may  appear  somewhat  paradoxical,  or  similar 
to  Maclaurin's  explanation  of  what  would  be  the  various  move- 
ments of  the  planets  were  they  operated  upon  by  various  laws  of 
gravity  which  have  no  existence  in  nature.  Nevertheless,  by  all 
philosophy,  it  requires  quadruple  the  force  of  attraction  to  bal- 
ance duplicate  the  motion  ;  and  the  motion  of  a  planet  is  doubled 
at  one  fourth  the  distance,  for  the  reason  that  the  force  is  quad- 
rupled. If,  then,  the  Newtonian  law  of  gravity  gave  but  just 
one  half  the  force  requisite  for  the  motion  of  the  moon,  or  of  the 
moon's  apogee,  as  found  by  Clairaut,  Euler  and  others,  then  in 
order  that  the  Newtonian  law  might  give  the  actual  or  observed 
motion,  it  would  be  necessary  for  the  moon  to  revolve  at  one 
fourth  the  distance  from  the  earth  that  it  now  does.  And  the 
like  in  respect  to  any  planet  in  the  solar  system ;  the  motion  be- 
ing just  double  that  which  would  be  sustained  by  the  Newtonian 
law  of  gravity  at  the  distance  at  which  the  planet  revolves.  And 
in  order  that  it  might  revolve  about  the  sun  retaining  the  mo- 
tion it  now  has,  it  must  revolve  at  one  fourth  the  distance  it  now 
does,  in  order  to  be  sustained  in  its  orbit  by  the  Newtonian  law 
of  gravity. 

In  respect  to  a  body  commencing  its  fall  towards  the  earth, 
as  there  is  no  portion  of  the  fall,  from  its  commencement,  how- 
ever small,  in  which  the  fall  or  motion  is  not  accelerated,  it  fol- 
lows as  a  consequence  to  such  axiom  or  proposition,  that  at  any 
given  point  of  space  of  the  fall,  or  at  any  given  instant  of  time 
during  the  fall,  the  amount  of  space  then  fallen  over  is  the  square 


APPENDIX.  289 

of  the  amount  of  force  then  expended  on  the  body  during  the 
fall,  and  consequently  that  the  intensity  of  the  force  varies  inverse- 
ly as  the  distance  varies. 

Nothing  could  be  devised  to  show  more  clearly  the  truth  which 
I  allege,  than  the  simplest  formula  that  can  be  constructed  for 
the  purpose  of  exhibiting  the  numerical  law  of  falling  bodies. 
If  we  conceive  a  body  to  fall  from  a  state  of  rest  over  a  given 
space  in  a  given  time,  (as  for  instance,  over  one  rod  in  one  sec- 
ond of  time,)  we  may  denote  the  operation  by 

1 

the  space  fallen  over,  (denoted  by  the  lower  figure  1,)  being  a 
unit  of  space,  and  its  square  root  denoted  by  the  upper  figure  1, 
will  as  well  denote  the  unit  of  force  employed  during  the  fall 
over  the  unit  of  space,  as  it  will  the  unit  of  time  in  which  the 
body  is  falling  over  the  unit  of  space. 

If  we  now  conceive  the  unit  of  space  to  be  divided  into  four 
equal  parts,  the  formula  will  be  thus, 

1       1 

123 

in  which  case  the  odd  numbers  in  the  lower  series  will  denote 
the  four  equal  parts  into  which  the  unit  of  space  is  divided.  If 
then  we  extract  the  square  root  of  the  lower  series,  taking  figure 
1  for  the  first  period  in  the  process  of  extraction,  the  square  root 
of  the  lower  series  will  be  expressed  by  figure  1  over  each  odd 
number  of  the  lower  series ;  which  square  root  expressed  by  the 
upper  series,  will  denote  as  well  a  division  of  the  unit  of  force, 
as  a  division  of  the  unit  of  time,  into  equal  parts.  Hence,  while 
the  body  is  falling  over  that  portion  of  the  unit  of  space  denoted 
by  figure  1  in  the  lower  series,  one  half  of  the  unit  of  force,  as 
also  one  half  of  the  unit  of  time,  will  be  expended  ;  and  the  re- 
maining half  of  the  unit  of  force,  and  also  of  the  unit  of  time 
will  be  expended  while  the  body  is  falling  over  three  fourths  of 
the  unit  of  space,  namely,  that  denoted  by  figure  3. 

So  the  unit  of  space  may  be  divided  into  as  many  equal  parts 
as  we  please,  even  to  an  infinite  or  indefinite  number  ;  in  which 
case  the  lower  series  of  the"  formula  will  be  expressed  by  the  nat- 
ural numbers  in  their  order,  while  the  equal  parts  of  such  unity 
of  space  will  be  denoted  by  the  odd  numbers  of  the  series  ;  and 
a  like  extraction  of  the  square  root  of  the  series,  (taking  figure  1 
at  the  left  as  the  first  period  in  the  process  of  extraction,)  will 
give  figure  1  over  each  odd  number,  as  the  square  root  of  the 
series ;  in  which  case  the  unit  of  force,  as  also  the  unit  of  time, 
will  be  divided  into  as  many  equal  parts  as  there  are  figures  in 
such  root.  And  it  is  manifest  that  however  far  we  extend  such 
37 


290  APPENDIX. 

division,  at  any  instant  or  point  of  the  fall, —  no  matter  how  little 
space  has  been  described  or  fallen  over,  —  the  sum  of  the  odd 
numbers  of  the  lower  series  described  by  the  fall  thus  far,  will  be 
the  square  of  the  number  of  equal  moments,  or  of  the  equal 
amounts  of  force  which  have  been  expended  thus  far.  Hence 
the  space  passed  over  or  described  in  the  fall  is  as  well  the 
square  of  the  amount  of  force,  as  of  the  amount  of  time  expend- 
ed ;  and  consequently  there  is  no  space  described  or  passed  over, 
without  acceleration. 

Such,  then,  enables  us,  in  the  most  simple  and  easy  manner, 
to  determine  the  law  of  gravity,  namely,  by  comparing  the  de- 
scent of  two  bodies  falling  from  a  state  of  rest,  but  from  different 
distances  from  the  same  centre  of  gravity. 

SUGGESTIONS    IN    REGARD    TO    CYLINDRICAL    SECTIONS. 

I  will  here  only  suggest  in  respect  to  the  three  principal 
planes  that  may  be  cut  from  the  prime  cylinder,  or  cylinder,  the 
diameter  of  whose  inscribed  sphere  is  unity  or  1. 

If  we  conceive  the  prime  cylinder  standing  upon  one  end, 
upon  a  horizontal  plane,  the  base  will  denote  the  prime  circle, 
and  if  such  cylinder  be  cut.  horizontally  through,  the  plane  will 
be  the  prime  circle ;  and  if  a  plane  be  cut  perpendicularly 
through  the  centre  of  the  cylinder,  the  plane  will  be  the  prime 
square. 

If  the  cutting  plane  commence  on  the  side  of  the  cylinder,  at 
any  point  between  the  base  and  altitude,  and  terminate  on 
the  opposite  side,  at  the  point  touching  the  periphery  of  the  base, 
the  plane  will  be  an  ellipse  (or  elongated  circle,)  the  area  of 
which  will  be  the  product  of  the  longest  by  the  shortest  diameter, 
multiplied  by  the  area,  or  one  fourth  of  the  circumference  of 
the  prime  circle.  The  longest  diameter  of  such  ellipse  may  be 
said  to  have  its  upper  and  lower  ends ;  the  upper  end  being  the 
one  above  the  base  of  the  cylinder. 

As  the  shortest  diameter  of  any  ellipse  cut  from  the  prime 
cylinder  is  always  unity,  it  is  therefore  manifest  that  the  area  of 
any  such  ellipse  is  to  the  longest  diameter,  as  Ma  of  the  circle 
is  to  1,  or  as  1  is  to  d a  or  b  a  of  the  circle  ;  and  hence,  that  the 
area  will  increase  or  decrease  by  the  same  ratio  that  the  longest 
diameter  will ;  the  consequence  is,  that  the  longest  diameter  (or 
major  axis)  of  any  ellipse  cut  from  the  prime  cylinder,  multiplied 
by  the  area  of  the  prime  circle,  gives  the  area  of  such  ellipse ; 
consequently,  if  the  major  axis  of  such  ellipse  be  =  da  or  ba 
of  the  circle,  the  area  of  such  ellipse  will  be  1,  or  is  =  m  of  the 
circle  whose  area  is  =  d  or  b  of  the  circle.  Thus,  the  area  of 


APPENDIX.  291 

any  ellipse  cut  from  the  prime  cylinder,  is  =  m  of  a  circle  whose 
b  or  d  is  =  the  major  axis  of  such  ellipse ;  and  hence,  if  the 
major  axis  be  at  its  maximum,  namely,  the  square  root  of  2,  the 
area  of  such  ellipse  is  necessarily  a  mean  between  Ma  and 
twice  Ma  of  the  circle  ;  and  such  ellipse  may  properly  be  termed 
the  prime  ellipse,  the  area  of  its  circumscribed  circle,  or  of  any 
circumscribed  parallelogram,  being  double  that  of  a  like  inscribed 
figure.  The  plane  of  such  ellipse  as  situated  in  the  cylinder, 
forms  an  angle  of  45°  with  the  base ;  and  as  the  focal  distance 
of  any  ellipse  cut  from  the  prime  cylinder  is  =  the  altitude  of 
the  upper  end  of  the  longest  diameter,  or  major  axis,  hence, 
the  focal  distance  of  the  prime  ellipse  is  unity.  And  as  the 
length  of  the  describing  thread  of  any  ellipse  is  =  the  major 
axis,  plus  the  focal  distance,  hence  the  length  of  the  describing 
thread  of  the  prime  ellipse  is  =  the  sum  of  the  longest  and 
shortest  diameters,  or  of  the  major  and  minor  axes.  Thus,  while 
the  major  axis  of  the  prime  ellipse  is  =  ba  8,  the  length  of  the 
describing  thread,  is  twice  d  a  8,  consequently,  from  one  end  of 
the  prime  ellipse,  to  the  nearest  focus,  is  =  v  a  8  —  for  in  respect 
to  any  p,  da  plus  v  a  is  =  b  a. 

So  the  area  of  the  prime  ellipse  is  =  b  or  d  of  the  circle,  whose 
area  is  1  — or  is  =  b  of  the  p  8  whose  area  is  the  reciprocal  of 
b  —  or  it  is  a  mean  between  the  area  of  the  square  whose  cir- 
cumference is  4,  and  the  area  of  the  circle  whose  circumference 
is  4. 

But  it  is  not  my  intention  here,  to  treat  of  the  qualities  of 
ellipses ;  nor  to  do  more  than  to  suggest  some  unity  of  purpose 
in  their  consideration,  which  is  certainly  to  be  desired  ;  if,  (as  is 
generally  supposed,)  planets  can  revolve  in  ellipses,  however  ec- 
centric—  or  in  ellipses  of  extreme  eccentricity,  and  in  which, 
(according  to  the  modern  version  of  Kepler's  law)  "the  squares 
of  the  periods  are  as  the  cubes  of  their  semi-major  axes."  Now, 
according  to  this  version,  if  we  conceive  two  planets  to  revolve 
around  the  same  central  force  in  orbits,  the  respective  circumfer- 
ences of  which  are  4,  one  orbit  being  a  circle  (either  centric  or 
eccentric,)  and  the  other  an  ellipse  of  extreme  ellipticity.  In 
such  case,  the  time  of  the  period  of  the  planet  revolving  in  the 
ellipse  must  be  1,  while  that  in  the  circle  will  be  .5.  Hence,  in 
the  ellipse,  the  time  of  the  period  will  have  no  regard  to  the 
space,  or  length  of  road  passed  over,  or  to  the  area  described  by 
the  radius  vector,  or  even  to  the  amount  of  force  of  attraction 
expended  on  the  planet  during  the  period.  And  if  we  conceive 
a  planet  to  revolve  in  an  ellipse,  we  must  conceive  it  to  revolve 
without  obeying  any  of  the  requirements  of  the  great  physical 
law  of  the  universe. 


., 

* 

292  APPENDIX. 

But  in  respect  to  the  circular  orbit,  (whether  centric  or  eccen- 
tric,) by  any  system  of  the  quadrature,  if  the  diameter  of  the 
orbit  be  =  b  a  of  any  given  j»,  the  time  of  the  period  will  be  the 
cube  or  third  power  of  B  of  a  like  p.  So  also,  if  the  diameter 
is  =  da  of  any  given  jt?,  the  time  of  the  period  is  the  cube  or 
third  power  of  D  of  a  like  p.  So  when  the  diameter  is  equal 
da  or  b  a  of  the  circle,  the  period  is  the  cube,  or  third  power  of 
B  or  of  D  of  the  circle. 

Hence,  the  reciprocal  of  A  of  any  given  p,  is  the  diameter  of 
an  orbit  in  which  the  period  is  the  third  power  of  B  of  a  like  p. 
And  the  reciprocal  of  M  a  of  any  />,  is  the  diameter  of  an  orbit 
in  which  the  time  of  the  period  is  the  third  power  of  D  of  a 
like  j9,  by  any  system  of  the  quadrature. 

Hence,  the  time  of  the  period  of  a  planet,  is  the  third  power 
of  the  square  root  of  radius  of  the  orbit. 

So  if  the  diameter  of  an  orbit  be  =  m  of  the  circle,  whose 
area  is  2, —  namely,  =  b  of  the  equal  square,  or  square  whose 
area  is  =  to  the  area  of  the  prime  circle,  or  whose  diameter  is 
one  eighth  of  the  square  of  the  circumference  of  the  prime  cir- 
cle, the  time  of  the  period  in  such  orbit  will  be  the  sixth  power 
of  m,  or  the  third  power  of  the  area  of  such  equal  square. 


• 


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NOV.  9  1940  M, 
IP/ 9    1974 


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